arXiv:1310.1778v1 [math-ph] 7 Oct 2013

Ludwig-Maximilians-Universität München

Diplomarbeit

Time Evolution in the external field problem

of Quantum Electrodynamics

Dustin Lazarovici

betreut von Prof. Dr. Peter Pickl,Mathematisches Institut der Ludwig-Maximilians-Universität

München

- Zur Erlangung des Grades eines Diplom-Mathematikers -

München, den 30. September 2011

Überarbeitete Version vom 19. Oktober 2011.

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Ludwig-Maximilians-Universität München

Diploma Thesis

Time Evolution in the external field problem

of Quantum Electrodynamics

Dustin Lazarovici

under supervision of Prof. Dr. Peter Pickl,Mathematisches Institut der Ludwig-Maximilians-Universität

München

- For the achievement of the degree : Diploma in Mathematics -

Submitted: September 30th, 2011

Time Evolution in the external field problemof Quantum Electrodynamics

Dustin Lazarovici ∗

Abstract

A general problem of quantum field theories is the fact that the free vacuum and thevacuum for an interacting theory belong to different, non-equivalent representations of thecanonical (anti-)commutation relations. In the external field problem of QED, we encounterthis problem in the form that the Dirac time evolution for an external field with non-vanishingmagnetic components will not satisfy the Shale-Stinespring condition, known to be necessaryand sufficient for the existence of an implementation on the fermionic Fock space. Therefore,a second quantization of the time evolution in the usual way is impossible.

In this work, we will present several rigorous approaches to QED in time-dependent, ex-ternal fields and analyze in what sense a time evolution can exist in the second quantizedtheory. It is shown that the construction of the fermionic Fock space as an “infinite wedgespace”, introduced in [DeDuMeScho], is equivalent to the more established construction of theFock space as the space of holomorphic sections in the dual of the determinant bundle over theinfinite-dimensional Grassmann manifold (see e.g. [PreSe]).

Concerning the problem of the time evolution, we first give a comprehensive presentationof the solutions proposed by Deckert et. al. in [DeDuMeScho], where the time evolution isrealized as unitary transformations between time-varying Fock spaces, and by Langmann andMickelsson in [LaMi96], where the authors construct a “renormalization” for the time evolutionand present a method to fix the phase of the second quantized scattering operator by paralleltransport in the principle fibre bundle GLres(H)→ GLres(H).

We provide a systematic treatment of the “renormalizations” introduced in [LaMi96] andshow how they can be used to translate between the second quantization procedure on time-varying Fock spaces and the second quantization of the renormalized time evolution. Weargue that non-uniqueness of the renormalizations corresponds to an additional freedom inthe construction of the second quantized S-operator in [LaMi96] which is expressed by theholonomy of the of the principle bundle Ures(H) and might no have been fully appreciated inthe original paper.

We provide rigorous proof for the fact that the second quantization by parallel transportpreserves causality. These findings seem to refute claims made in [Scha] that the phase of thesecond quantized S-matrix is essentially determined by the requirement of causality. The resultcan also be applied in the context of time-varying, showing that the implementations can bechosen such that the semi-group structure of the time evolution is preserved.

We propose a simple solution to the problem of gauge anomalies in the procedure of Lang-mann and Mickelsson, showing that the second quantization of the scattering operator can bemade gauge-invariant by using a suitable class of renormalizations.

∗Mathematisches Institut, LMU München. [email protected]

i

Contents

Preface 1

1 Introduction 51.1 The Dirac Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 An Intuitive Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Dirac Sea versus Electron-Positron Picture . . . . . . . . . . . . . . . . . . . . 10

2 Polarization Classes and the Restricted Transformation Groups 132.1 Polarization Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The restricted Unitary and General Linear Group . . . . . . . . . . . . . . . . 16

2.2.1 Lie Group structure of GLres(H) and Ures(H) . . . . . . . . . . . . . . 172.3 The restricted Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Projective Representations and Central Extensions 213.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Central Extensions of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Projective Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 Cocycles and Cohomology Group . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Central Extensions of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 The Central Extensions GLres(H) & Ures(H) 354.1 The Central Extension of GLres . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 The complete GLres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.2 Lie Group- and Bundle- structure . . . . . . . . . . . . . . . . . . . . . 394.1.3 The Central Extension of Ures and its Local Trivialization . . . . . . . 41

4.2 Non-Triviality of the Central Extensions . . . . . . . . . . . . . . . . . . . . . 43

5 Three Routes to the Fock space 475.1 CAR Algebras and Representations . . . . . . . . . . . . . . . . . . . . . . . . 47

5.1.1 The Field Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1.2 C∗ - and CAR- Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 575.1.3 Representations of the CAR Algebra . . . . . . . . . . . . . . . . . . . 58

5.2 The Infinite Wedge Space Construction . . . . . . . . . . . . . . . . . . . . . . 615.3 The Geometric Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Equivalence of Fock Space Constructions . . . . . . . . . . . . . . . . . . . . . 775.5 CAR Representation on Fgeom . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Time-Varying Fock spaces 816.1 Identification of Polarization Classes . . . . . . . . . . . . . . . . . . . . . . . . 826.2 Second Quantization on Time-varying Fock Spaces . . . . . . . . . . . . . . . . 85

6.2.1 Particle Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.2.2 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

ii

7 The Parallel Transport ofLangmann & Mickelsson 917.1 The Langmann-Mickelsson Connection . . . . . . . . . . . . . . . . . . . . . . 92

7.1.1 The Maurer-Cartan Forms . . . . . . . . . . . . . . . . . . . . . . . . . 927.1.2 Connection Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947.1.3 Local Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.2 Parallel Transport in the GLres(H)-bundle . . . . . . . . . . . . . . . . . . . . 977.3 Classification of Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8 Geometric Second Quantization 1018.1 Renormalization of the Time Evolution . . . . . . . . . . . . . . . . . . . . . . 102

8.1.1 Renormalization of the Hamiltonians . . . . . . . . . . . . . . . . . . . 1068.1.2 Notes on Renormalizations . . . . . . . . . . . . . . . . . . . . . . . . . 109

8.2 Geometric Second Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . 1128.2.1 Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.3 Holonomy of the Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1208.4 Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

9 Closing Arguments 127

Appendix 133A.1 Commensurability and Polarization Classes . . . . . . . . . . . . . . . . . . . . 133A.2 On Banach Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134A.3 Derivation of the Charge Conjugation . . . . . . . . . . . . . . . . . . . . . . . 136A.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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NOTATION

Notation and Mathematical Preliminaries

Throughout this work we use “natural units” in which ~ = c = 1.We use the Minkowski metric with signature (+,−,−,−).

Furthermore, we introduce the following notations:

H a separable, complex Hilbert space.

B(H) the space of bounded operators on H.

GL(H) the space of bounded automorphisms of H.

U(H) the group of unitary automorphisms of H.

Ip(H,H′) the p-th Schatten class of linear operators T : H → H′ for which

(‖T‖p)p := tr[(T ∗T )p/2] <∞.

‖·‖p is a norm that makes Ip(H,H′) a Banach space. It satisfies‖AT‖p ≤ ‖A‖ ‖T‖p and ‖TB‖p ≤ ‖B‖ ‖T‖p, forA ∈ B(H′), B ∈ B(H). Thus, Ip(H,H) =:

Ip(H) is a two-sided ideal in the algebra of bounded operators on H.If T =

∑k≥0

µk|fk〉〈ek| is a singular-value decomposition, the p-th Schatten norm corre-

sponds to the `p norm on the sequence (µk)k of singular values.If an operator T is in Ip(H,H′) for some p, then T is compact.

In particular,

I1(H) the ideal of trace-class operators for which tr(T ) :=∑k≥0

〈ek, T ek〉 is well-defined and

independent of the Hilbert basis (ek)k≥0 of H.

I2(H,H′) the class of Hilbert-Schmidt operators H → H′. The product of two Hilbert-Schmidtoperators is in the trace class with ‖ST‖1 ≤ ‖S‖2 ‖T‖2.

Id+ I1(H) = T = Id + A | A ∈ I1(H) the set of operators for which the Fredholmdeterminant is well defined by

det(1 +A) :=

∞∑k=0

tr(∧k

(A))

=

∞∑k=0

∑i1<···<ik

λi1 · · ·λik ,

for (λn)n, the eigenvalues of A. The Fredholm determinant has the following properties:

i) |det(1 +A)| ≤ exp(‖A‖1).

ii) (1 +A) is invertible, if and only if det(1 +A) 6= 0.

iii) det(S) det(T ) = det(ST ), for S, T ∈ Id+ I1(H).

iv) If (ek)k≥0 is on ONB of H, then det(T ) = limN→∞

det(〈ei, T ej〉

)i,j≤N

GL1(H) := GL(H) ∩(Id + I1(H)

)denotes the set of bdd. isomorphism with well-defined

Frdholm determinant.

U1(H) := U(H)∩(Id+ I1(H)

)denotes the set of unitary operators with well-defined Frdholm

determinant.

All further notations will be introduced in the course of the work.

iv

Preface

Quantum Electrodynamics, or short: QED, is widely considered to be the most successfultheory in entire physics. Indeed, its predictions have been confirmed time and time again withremarkable precision by various experiments in particle accelerators and laboratories all overthe world. Probably the most famous and most spectacular demonstration of the potency ofQuantum Electrodynamics is the prediction of the anomalous magnetic moment of the elec-tron, known as “g - 2” in the physical literature. This electron g-factor has been measured withan accuracy of 7.6 parts in 1013, i.e. with a stupendous precision of 12 decimal places andfound to be in full agreement with the theoretical prediction (Odom et.al., Phys. Rev. Lett.97, 030801 (2006)). Actually, we have to be more precise: since the fine structure constant αenters every QED-calculation as a free parameter, we have to gauge it by other experiments or,equivalently, express every QED-measurement as an independent measurement of α. In thissense, theory and experiment are in agreement up to 0.37 parts per billion i.e. to 10 decimalplaces in the determination of α (Hanneke et.al., Phys. Rev. Lett. 100, 120801 (2008)). Thishas often been called the best prediction in physics and whether this is adequate or not, it iscertainly very impressive.In one of the standard textbooks on quantum field theory it is even said that “Quantum Electro-dynamics (QED) is perhaps the best fundamental physical theory we have”. (Peskin, Schröder,“An Introduction to Quantum Field Theory”, 1995 ).

In this light, it might seem surprising that many of the brilliant minds that actually developedthe theory were not quite as enthusiastic about it. In a talk given in 1975, P.A.M. Diracfamously expressed:

“Most physicists are very satisfied with the situation. They say, Quantum Electro-dynamics is a good theory, and we do not have to worry about it any more. I mustsay that I am very dissatisfied with the situation, because this so-called good theorydoes involve neglecting infinities which appear in its equations, neglecting them inan arbitrary way. This is just not sensible mathematics. Sensible mathematicsinvolves neglecting a quantity when it turns out to be small - not neglecting it justbecause it is infinitely great and you do not want it!” 2

Cited after: H. Kragh, Dirac: A scientific biography, CUP 1990

And even in his Nobel lecture, where the occasion would have excused some enthusiasm,Richard Feynman said:

2What I find most remarkable about this quote is that it takes such a great scientist to statesomething so obvious and be taken seriously.

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PREFACE

“That is, I believe there is really no satisfactory quantum electrodynamics, but I’mnot sure. [...] I don’t think we have a completely satisfactory relativistic quantum-mechanical model, even one that doesn’t agree with nature, but, at least, agrees withthe logic that the sum of probability of all alternatives has to be 100%. Therefore,I think that the renormalization theory is simply a way to sweep the difficultiesof the divergences of electrodynamics under the rug. I am, of course, not sure ofthat.”

R.P. Feynman: "The Development of the Space-Time View of QuantumElectrodynamics", Nobel Lecture (1965)3

Comparing these two statements with the assessment of Peskin and Schröder and really withthe general spirit of the scientific community of today, one might think that great progress hasbeen made on the foundations of QED ever since. Quite frankly, I don’t see where. Of course,experimental success has steadily strengthened our trust in the usefulness of the frameworkof quantum field theory, but I don’t think this was really the concern expressed by Feynmanand Dirac. Over the years, we might have become desensitized to the problems of QED, butit seems to me that we haven’t done a very good job at fixing them. So, what then is wrongwith QED?

For once, more then half a century after its development, Quantum Electrodynamics stilllacks a rigorous mathematical formulation. It is well known that QED (just as all realisticquantum field theories) relies on different “renormalization” schemes to render its predictions(more or less) finite. And even after renormalization, the S-matrix expansion is widely believedto have zero radius of convergence in the coupling constant α. No matter how crafty physicistshave gotten at manipulating infinities, this fact remains highly unsatisfying from a mathemat-ical point of view. It is also well known that every attempt to axiomatize (3+1 dimensional)QED, i.e. every attempt to base the theory on a consistent set of formal requirements, hasfailed – and is bound to fail for any “serious” quantum field theory. Interestingly enough, thisrealization had not so much shaken confidence in the theories themselves but rather ended theprogram of axiomatizing fundamental physics.

A fact widely ignored by physicist is that the mathematical deficiencies of QED (and otherfield theories) do not begin at the computational level –they are really much more basic. Theusual formulations of the theory are intrinsically ill-defined and it’s not every clear how towrite things down in a way that is mathematically meaningful. Most physicists seem eithernot to know or not to care about these kind of problems. This has created a somewhat tragi-comical situation. Nowadays, very little work is done on the rotten foundations of the theory.Physicists don’t work on it, because they believe the theory is so good that there’s nothing leftto do. Mathematicians don’t work on it, because they find the theory so bad that they don’teven know where to start.

But apart from all mathematical problems, QED (and really the entire Standard Model ofparticle physics) is incomplete in a very different sense: it’s lacking an “ontology”, a meaningfulinterpretation of the mathematical framework, providing a clear and coherent picture of whatthe theory is really about. I believe that a fundamental theory of nature has to clearly identifythe basic elements of its physical description (the local beables, in the sense of John Bell), which

3http://nobelprize.org/nobel_prizes/physics/laureates/1965/feynman-lecture.html

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PREFACE

we can think of as the basic constituents of physical “reality”. And it has to tell us, how ex-actly these physical objects correspond to abstract objects or parameters in the mathematicalformulation of the theory; indeed, we should insist that this correspondence be as simple andimmediate as possible.Thinking about Quantum Electrodynamics, just try in all honesty toanswer the question: what is the theory actually about? What are the fundamental physicalobjects of its description? Is QED essentially a theory about charged particles? This is prettymuch the way we use to think and talk about it; but if you inquire a little further, a particlephysicist will have to tell you that this is not “really” what’s going on. And indeed, taking alook into any textbook on quantum field theory or the Standard Model, we’re going to findplenty of “particles”, neatly listed in various tables or swirling around in funny little diagrams– yet, there aren’t actually any particles in the theory.4 So, maybe QED is a theory about“quantum fields”? The expression “quantum field theory” might suggest that kind, but theanswer is not too convincing, as the role of the “fields” in the theory remains rather obscure.Mostly, they seem to appear as a formal device for setting up the perturbation expansions or forderiving the equations of motion from a principle of least action. I might be wrong about that.Even then, however, explaining what the quantum fields are actually supposed to be and howthey constitute the physical world that we live in, seems like a formidable task and not manypeople, who invoke this answer, like to engage in it. Maybe QED is merely about “transitionamplitudes”. To my understanding, this very pragmatic standpoint was advocated by suchdistinguished scientists as Werner Heisenberg, and it might very well be logically consistent,although I think it requires some mental gymnastics to avoid questions like what transitionsfrom what into what?

One might call such concerns “metaphysical”, but to me, they are as physical as it gets.And with such considerations in mind, the state of modern physics in general and of QuantumElectrodynamics in particular seems pretty bad. I have to repeat, though, that the vast major-ity of physicists does not share this kind of pessimism. The reasons for this may be historicalor sociological or they might, of course, lie in my incomplete understanding of things. Thereare various possible reasons for this and a detailed discussion would certainly go beyond thescope of this introduction. However, I like the irony in the idea that the very genius of RichardFeynman might have to take some of the blame. His ingenious method of visualizing the formalexpansions of quantum field theories by means of the famous diagrams that carry its name,provided us with most of the intuition we have for the physical processes in particle physicsand coined the way we use to think and talk about the theory. We are so used to talkingabout particles scattering from each other, about photons being emitted and absorbed or pairsof virtual particles “screening" charges and so on, that we tend to forget that none of this isactually in the theory. As my first teacher of quantum field theory used to say: “it’s just anice, cartoonish way to talk about these things.”

Above, I have suggested that physicists don’t work on foundations of QED anymore, be-cause they don’t see any problems with the theory. This is just partly true. Actually, manyphysicists do acknowledge that the theory is deeply flawed but have – de facto – given up on it.Instead, they have adopted the point of view that QED (or rather the entire Standard Model)is indeed not a fundamental theory of nature, but merely a low energy approximation to a

4Indeed, in the Standard Model, particles can have mass, charge, spin, even color, flavor or families,but no location in space-time. And therefore no substance as a physical object.

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PREFACE

fundamental theory (maybe a “theory of everything”) still waiting to be discovered. All theproblems we’re facing today, that is the hope, will vanish, once said theory is found. StringTheory is usually considered to be the best candidate for such a fundamental theory of na-ture. Personally, I am sceptical whether the results of String Theory after 20 years of intensiveresearch justify this kind of optimism, but that’s a different debate. Anyways, claiming thatQED will have to wait for the next big scientific revolution to resolve its various issues mightbe a valid standpoint. I just have two objections that I have to mention.

First: it has never been like that in the history of physics. Classical mechanics wereperfectly well defined and well understood before Special Relativity and Quantum Mechanicscame to extend the picture. The Maxwell-Lorentz theory of electromagnetism is a beautifultheory, both physically and mathematically, except for one little detail: the electron self-interaction. This problem was not solved but inherited by the presumingly “more fundamental”theory of Quantum Electrodynamics, where it made quite a prominent career under the nameof “ultraviolet divergence”.5

My second objection is this: even if the final answers do lie beyond QED, isn’t it stillimportant to understand as well as possible what exactly goes wrong and what can and cannotbe done? Isn’t it possible, even likely, that insights of this kind will lead the way to a new,better behaved, maybe more fundamental description? As John Bell put it 6:

“Suppose that when formulation beyond FAPP [for all practical purposes] is at-tempted, we find an unmovable finger obstinately pointing outside the subject, tothe mind of the observer, to the Hindu scriptures, to God, or even only Gravita-tion? Would not that be very, very interesting?”

J. Bell, “Against Measurement”, in: [Bell]

It is in this spirit that I wrote this thesis and that the research program started by myteachers and colleagues should be understood. The goal is ambitious, yet humble at the sametime. We do not expect to “fix” QED, or solve all the problems that have troubled so manygreater physicists before us. However, we hope to get a better understanding of the difficulties,approach them in a systematic way and see how far one can get with rigorous mathematics.The work I am presenting here is of rather technical nature and mostly concerned with thetask of lifting the unitary time evolution to the fermionic Fock space (“second quantization”)in the exterior field problem of QED. I hope that this will provide some insights into thefundamental difficulties of QED in particular and of relativistic quantum theory in general. Tome, at least, it was a humbling realization to learn at what basic level the formalism alreadyfails. Ultimately, though, my personal believe is that if we want to make significant progresstowards a meaningful, well-defined, fundamental theory we need to think the formal aspectsand the conceptual aspects together and return to a way of doing physics that takes both,rigorous mathematics and true physical understanding seriously.

5Above all, to this problem, QED added the “infrared divergence”, so things really just got worseon that front.

6Actually, Bell wrote this about non-relativistic Quantum Mechanics, but his appeal seems evenmore urgent in the context of modern quantum field theories

4

Chapter 1

Introduction

1.1 The Dirac Equation

We begin our study of the external field problem in Quantum Electrodynamics with the one-particle Hilbert space H := L2(R3,C4) of square-integrable C4-valued functions. The funda-mental equation of motion is the famous Dirac equation

(i∂/−m)Ψ(t) = (iγµ∂µ −m)Ψ(t) = 0 (1.1.1)

where Ψ(t) ∈ L2(R3,C4) for every fixed t ∈ R. The gamma matrices γ0, γ1, γ2, γ3 form a4-dimensional complex representation of the Clifford algebra Cl(1, 3), i.e. they satisfyγµ, γν = 2gµν · 1 where gµν is the Minkowski metric tensor.Using (γ0)2 = 1, we can rewrite the Dirac equation in Hamiltonian form as

i ∂t Ψ = D0 Ψ :=(−iα · ∇+mβ

)Ψ (1.1.2)

Here, the notations β = γ0 and αµ = γ0γµ are common.In the presence of an electromagnetic field described by a vector potentialA = (Aµ)µ=0,1,2,3 = (Φ,−A), the partial derivative in the Dirac equation (1.1.1) is replaced bythe covariant derivative ∂µ + ieAµ, adding to the Hamiltonian the interaction potential

V (t) = e αµAµ = −e α · A + eΦ (1.1.3)

As a side-note we remark that a mathematically more sophisticated description would startwith a space-time manifold M ×R, where M is a (3-dimensional) compact manifold with spin-structure, and realize H as the space of L2-sections in a spinor-bundle overM (cf. [LawMich]).

It is well known that the free Dirac Hamiltonian D0 is unstable. It has the continuousspectrum (−∞,−m] ∪ [+m,+∞) which gives rise to a splitting of the one-particle Hilbertspace H = L2(R3,C4) into two spectral subspaces H = H+ ⊕ H−. Physical interpretationof the negative energy free states is difficult and has troubled physicists for many years. Inparticular, as the Hamiltonian is unbounded from below, it would be possible to extract anarbitrary amount of energy from the system, which is unphysical. To deal with this problems,P.A.M. Dirac proposed the so-called Dirac sea- or hole theory :

5

1.1. THE DIRAC EQUATION

“Admettons que dans l’Univers tel que nous le connaissons, les états d’energienégative soient presque tous occupés par des électrons, et que la distribution ainsiobtenue ne soit pas accessible à notre observation à cause de son uniformité danstoute l’etendue de l’espace. Dans ces conditions, tout état d’energie négative nonoccupé représentant une rupture de cette uniformité, doit se révévler à observationcomme une sorte de lacune. Il es possible d’admettre que ces lacunes constituentles positrons.” P.A.M. Dirac: Théorie du Positron. [Dir34]

According to Dirac, the negative energy states are almost entirely occupied by an infinitenumber of electrons – the Dirac sea – which due to its homogeneous distribution is hiddenfrom physical observation. The Pauli exclusion principle will then work to prevent transitionof positive energy electrons to states of arbitrarily negative energy, so that the system isstable. “Holes” in the otherwise homogeneously filled Dirac sea will appear as particles ofpositive energy but opposite charge and represent what we call positrons. Transition of anelectron from the negative energy spectrum to the positive energy spectrum in the presence ofan electromagnetic field will look like the simultaneous creation of an electron and a positronto the outside observer. Conversely, when a positive energy electrons drops into an unoccupiedstate of negative energy, we see the annihilation of an electron/positron pair with energy beingemitted in form of radiation.

Figure 1.1: Dirac’s hole theory.

Despite the obvious peculiarities of the model – that is, the existence of an infinite number ofparticles whose status remains to discuss – it provides an ingenious picture explaining the mostimportant phenomena of relativistic quantum theory in a clear and elegant way. Admittedly,the Dirac sea is usually absent in “modern” presentations of QED, though it seems to methat it hasn’t been replaced by an equally compelling physical picture. Anyways, Dirac’stheory provides a good intuition for the difficulties of relativistic quantum theory and can beunderstood as the physical motivation behind most of the mathematically rigorous approachesto the second quantized theory that will be presented in this work.

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1.2. AN INTUITIVE APPROACH

1.2 An Intuitive ApproachFrom nonrelativistic Quantum Mechanics we are familiar with the fact that the n-fermionHilbert space is the n-fold exterior product

∧nH of the one-particle Hilbert space H. Thisspace is spanned by decomposable states of the form v1 ∧ · · · ∧ vn. The Hermitian scalarproduct is given by:

〈v1 ∧ · · · ∧ vn, w1 ∧ · · · ∧ wn〉 = det(〈vi, wj〉)i,j (1.2.1)

Under a unitary time evolution U = U(t1, t0), the states evolve in the obvious way:

v1 ∧ · · · ∧ vnU(t1,t0)−−−−−→ Uv1 ∧ · · · ∧ Uvn (1.2.2)

Projectively (i.e. mod C), such states are in one-to-one correspondence with n-dimensionalsubspaces of the Hilbert space H by

v1 ∧ · · · ∧ vn 7−→ span(v1, . . . , vn) =: V ⊂ H, (1.2.3)

for if we take w1, . . . , wn ∈ H with span(w1, . . . , wn) = span(v1, . . . vn), i.e. a different basis ofV, we find

w1 ∧ · · · ∧ wn = det(R) v1 ∧ · · · ∧ vn (1.2.4)

with R, the matrix in GLn(C) transforming the basis (v1, . . . , vn) into (w1, . . . , wn).(Note that if v1, . . . , vn are not linearly independent, then v1 ∧ · · · ∧ vn = 0).

In the setting of relativistic Quantum Electrodynamics, however, we have a Dirac seacontaining infinitely many particles which makes the situation more complicated. Projectivedecomposable states are now in correspondence with certain infinite-dimensional subspacesof H, so-called polarizations (see Def. 2.1.1). For example, in the unperturbed Dirac sea(the ground state), all negative energy states are occupied and all positive energy states areempty, so this configuration of the Dirac sea corresponds to the subspace V := H− ⊂ H.Under a unitary transformation U , a time evolution U = UA(t1, t0), let’s say, V evolves intoW := U(V ) = UV .

What does this tell us about the physics? First and foremost we would like to ask: “Howmany electrons and how many positrons (holes) were created?”. Thanks to Dirac’s ingeniouspicture, we have a very simple intuition of how to answer this question: we just count! Thenegative energy states that remain occupied correspond to the subspaceW∩H−. Consequently,the number of holes is just the codimension of W ∩ H− in H−, i.e. the dimension of thefactor space H−/(W ∩ H−). Similarly, the Dirac sea picture tells us that we can think of thedimensions of W complementary to W ∩ H− as being populated by electrons that have beenlifted from the sea to positive energies. In conclusion:

#electrons ≈ dim(W/(W ∩H−)

). (1.2.5)

#holes ≈ dim(H−/(W ∩H−)

). (1.2.6)

The net-charge of W is then:

dim(W/(W ∩H−)

)− dim

(H−/(W ∩H−)

). (1.2.7)

In order for all of this to make sense, we have to require that both (1.2.5) and (1.2.6) are finite.Such polarizations V and W satisfying

dim(W/(W ∩ V )

)<∞ and dim

(V/(W ∩ V )

)<∞ (1.2.8)

are called commensurable.

7


There is a natural generalization of commensurability (in fact corresponding to a closure in atopological sense): If PV and PW are the orthogonal projections onto V and W respectively,we require that

PV − PW is a Hilbert-Schmidt operator.

In this case, we will say that V and W belong to the same polarization class. At first glance,this looks nothing like the condition of commensurability formulated above. But note thatPV − PW being of Hilbert-Schmidt type just means that (PV − PW )∗(PV − PW ) is in thetrace-class, i.e. (since orthogonal projections are self-adjoint)

tr(PV − PWPV + PW − PV PW ) <∞. (1.2.9)

This is starting to look more like what we’re going for. Orthogonal projections are self-adjoint operators with eigenvalues 1 (on the respective subspace) and 0 (on the orthogonalcomplement). Thus, tr(PV ) = dim(V ) whenever this is finite. And if PV and PW commute(which they usually don’t), PV PW = PWPV = PV ∩W and so tr(PV − PWPV ) really countsthe dimension of the orthogonal complement of W ∩ V in V and so forth. Therefore, it doesindeed make sense to regard (1.2.9) as a generalization of (1.2.8).The precise relationship between (1.2.9) and (1.2.8) is discussed in Appendix A.1, but we hopethat at this point, the reader is convinced that polarization classes are an adequate concept.

Similarly, there’s an abstract generalization of (1.2.7) counting the net-charges of polar-izations : If V and W are in the same polarization class, then ind(PW |V→W ) is well definedand we call this number the relative charge of V and W . Again, the motivation becomes moreclear if we remember that

ind(PW |V→W ) = dim ker(PW |V→W ))− dim coker(PW |V→W )

= dim ker(PW |V→W ))− dim (W/PW (V ))

This coincides with (1.2.7), whenever the latter is well defined (see Appendix A.1).

Returning to our initial setup with V = H− and W = UV for the unitary transformationU ∈ U(H), we denote by P− the orthogonal projection onto H− and by P+ the orthogonalprojection onto H+. In particular, P− + P+ = 1. As W results from H− by the unitarytransformation U , the orthogonal projection is PW = UP−U

∗. Therefore:

PV − PW Hilbert-Schmidt ⇐⇒ P− − UP−U∗ Hilbert-Schmidt⇐⇒ P−U − UP− Hilbert-Schmidt⇐⇒ P−U(P− + P+)− (P− + P+)UP−

= P−UP+ − P+UP− Hilbert-Schmidt.

As P−UP+ and P+UP− map from and into complementary subspaces, this is satisfied if andonly if both

U−+ := P−UP+|H+→H− and U+− := P+UP−|H−→H+(1.2.10)

are of Hilbert-Schmidt type. This is known as the Shale-Stinespring condition.

In the remainder of this work we will invoke more or less fancy mathematics to constructFock spaces and implement unitary transformations on them, but in the end, whatever pathwe take, it always comes down to this: the unitary time evolution would have to fulfill theShale-Stinespring condition. Otherwise, the Dirac sea becomes too stormy; the electromagneticfield creates infinitely many particles and we have no chance to compare initial and final statein a meaningful way. In other words: we cannot accommodate initial and final state in oneand the same Fock space. There is no way around this, at least not in the usual mathematicalframework.

8


It is obvious that we have stumbled upon a serious difficulty, but it might be less obvioushow bad the problem really is, because it is not evident how restrictive the Shale-Stinespringcriterion turns out to be. We might hope that in most – if not all – physically relevantsituations, things turn out to be sufficiently nice and everything is going to work out well. Butactually, we shouldn’t expect that. We should expect something like that, if the difficultieswere of a merely technical kind. But this is not the case. The Shale-Stinespring condition is nota petty mathematical prerequisite of the type “let f twice continuously differentiable”; muchrather, it reflects the problem of infinite particle creation which seems to be deeply inherentto relativistic quantum theory. Finally, all remaining hopes are destroyed by the followingtheorem:

Theorem 1.2.1 (Ruijsenaars, 1976).

Let A = (A0,−A) ∈ C∞c (R4,R4) be an external vector potential and UA(t, t′) the correspondingDirac time evolution on H. Then UA(t, t′) satisfies the Shale-Stinespring condition for all t, t′if and only if A = 0, i.e. if and only if the spatial part of the A-field vanishes identically.1

This and similar results seem not to receive the attention they deserve. Ruijsenaars theoremshows that the unitary time evolution can typically not be implemented on the fermionic Fockspace. In other words: there is no well-defined time evolution in Quantum Electrodynamics.

One might wonder why it’s only the spatial part of the vector potential (≈ the magnetic field)which causes the problem. The following considerations might provide some intuition:Recall that the free Dirac Hamiltonian, in momentum-space, is

D0(p) = α · p+ βm (1.2.11)

which reduces to D0(0) = βm for a particle at rest. In the so-called Dirac representation,

β =

(1 00 −1

).

Obviously, the standard basis vectors represent eigenstates with energy ±m.In the presence of an electromagnetic field with vector potential A = (Aµ)µ=0,1,2,3, there is anadditional interaction term

V A = e

3∑µ=0

αµAµ,

where α0 = 1 and Aµ is the Fourier transform of Aµ (acting as convolution operators). Theelectric potential Φ = A0 is harmless, it just shifts the energy of the particles by a finite amount.But the alpha matrices satisfy βαj = −αjβ for j = 1, 2, 3, which means that they are mappingnegative energy eigenstates of D0 to positive energy eigenstates (and vice-versa). Intuitivelyspeaking, the magnetic field “rotates” the Dirac sea into H+.

So far, we have only looked at the action of unitary transformations on polarizationswhich correspond to projective states. Given a unitary operator U that does satisfy the Shale-Stinespring condition, we encounter another difficulty when we try to define its action onnon-projective states: its lift to the Fock space is well-defined only up to a complex phase.This is a well-known result (for example from representation theory), but our intuitive consid-erations are good enough to see why this is the case.

1It might seem suspicious that the condition A ≡ 0 is blatantly not gauge-invariant. This, however,just points to another deficiency of QED. In contrast to common believe, the theory is not gauge-invariant in a naive sense. Gauge transformations as well do generally not satisfy the Shale-Stinespringcondition and therefore cannot be implemented as unitary transformations on the Fock space. We’llcome back to this problem in Chapters 6 and 7.

9

1.3. DIRAC SEA VERSUS ELECTRON-POSITRON PICTURE

Again we try to generalize the description of an n-fermion system represented in∧nH to

QED-states with an infinite number of particles. We may think of such a state as an infinitewedge product, i.e. a formal expression

Ψ = v0 ∧ v1 ∧ v2 ∧ v3 ∧ . . . ,

where the subspace spanned by the vj is in the polarization class of H−. Now let’s considerthe simplest case in which the vj are all eigenstates of the operator U with eigenvalues eiϕj .On the n-fermion state v0 ∧ v1 ∧ . . . ∧ vn, the operator U acts like

v0 ∧ v1 ∧ . . . ∧ vnU−−→ eiϕ0v0 ∧ eiϕ1v1 ∧ . . . ∧ eiϕnvn = e

i( n∑j=1

ϕj

) (v0 ∧ v1 ∧ . . . ∧ vn

)But again, things become more complicated in the infinite-dimensional case. On Ψ = v0 ∧ v1 ∧v2 ∧ . . . , the unitary transformation U would have to act like

v0 ∧ v1 ∧ v2 ∧ . . .U−−→ Uv0 ∧ Uv1 ∧ Uv2 ∧ . . . = eiϕ0v0 ∧ eiϕ1v1 ∧ eiϕ2v2 ∧ . . .

“ = “ ei(?) (v0 ∧ v1 ∧ v2 ∧ . . .).

The product of infinitely many phases does not converge in general and thus the phase of the“second quantization” of U , acting on the Fock space, is not well-defined. This U(1)-freedom isknown as the geometric phase in QED. “Geometric”, because we will identify it as the structuregroup of a principle fibre bundle over the Lie group of implementable unitary operators.

1.3 Dirac Sea versus Electron-Positron PictureConsidering the problems described so far, one might observe that all difficulties originate inthe fact that we are dealing with an infinite number of particles. Hence one might go on tosuggest that the problems might be solved by abandoning the Dirac sea with its infinitely manyelectrons and switching from the “electron-hole picture” to the “particle-antiparticle” picture, asis usually done in modern quantum field theory (indeed, most textbooks do suggest, implicitlyor explicitly, that this constitutes a significant improvement.) The first statement is, of course,correct. The second one is not. To see why this is so, let’s discuss what all this actually means.

What I’m referring to as the “particle-antiparticle-picture” or “electron-positron-picture”, asopposed to the Dirac sea - or electron-hole - picture, is a formulation in which the problematicnegative spectrum of the Dirac Hamiltonian is fixed in a rather ad hoc way by mapping negativeenergy solutions of the Dirac equation to positive energy solutions with opposite charge, whichare then interpreted as antiparticles (positrons). The Dirac sea is then omitted altogether aspart of the physical description. This reinterpretation is also reflected in the mathematicalformalism. In standard physics textbooks, this is usually done in a rather naive way :

The formal quantization of the Dirac field yields the expression

Ψ(x) =

∫d3p

(2π)3

1√2Ep

∑s

(aspus(p)e−ipx + bsp v

s(p)eipx) (1.3.1)

leading to the second quantized Hamiltonian

H =

∫d3p

(2π)3

∑s

(E(p) as∗p asp − E(p) bs∗p b

sp). (1.3.2)

This Hamiltonian is unstable, i.e. unbounded from below, as every particle of the type createdby b∗ decreases the total energy by at least its rest-mass m.

10


Thus, one uses the canonical commutation relations

brp, bs∗q = brpbs∗q + bs∗q b

rp = (2π)3δ3(p− q)δr,s (1.3.3)

to write −bs∗q bsp = +bspbs∗q − (2π)3 δ(0) and simply interchanges the roles of b and b∗.

After renaming the operators accordingly, one gets the stable Hamiltonian

H =

∫d3p

(2π)3

∑s

(E(p) as∗p asp + E(p) bs∗p b

sp) (1.3.4)

plus an infinite “vacuum energy” that physicists boldly get right of by “shifting the energy” or,in other words, ignoring it.

A less playful approach would use the charge conjugation operator C, an anti-unitary operatormapping negative energy solutions to positive energy solutions of the Dirac equation withopposite charge. Then construct the Fock space

F =∧H+ ⊗

∧C(H−) (1.3.5)

and define the field operator Ψ which is a complex anti-linear map into the space B(F) ofbounded operators on F . Explicitely,

Ψ : H → B(F), Ψ(f) = a(P+f) + b∗(P−f) (1.3.6)

where a is the annihilation operator on∧H+ and b∗ the creation operator on

∧C(H−)

(i.e. for g ∈ H−, b∗(g) creates the state Cg in∧C(H−) ).The idea is that

∧H+ contains the

electron-states and∧C(H−) the positron-states, all of positive energy. The vacuum state Ω is

just 1 = 1⊗1 ∈ F . If the reader excuses some physics jargon, we can say that the field operatorΨ “creates” antiparticles and “annihilates” particles. Correspondingly, the adjoint operator Ψ∗

“creates” particles and “annihilates” antiparticles.

This construction is carried out in more detail in the next chapter but clearly, as it involvesonly states of positive energy, it leads to a positive definite second quantized Hamiltonian on theFock space. Now, we can relate this construction to the Dirac sea description in the followingway: In the Dirac sea description, the vacuum state corresponds to the “unperturbed sea” inwhich all free, negative energy states are occupied by electrons and all positive energy statesare empty. Formally, we may think of this state as an “infinite-wedge product”. We pick abasis (ek)k∈Z of H, s.t. (ek)k≤0 is a basis of H− and (ek)k>0 a basis of H+, and represent thevacuum by the formal expression

Ω = e0 ∧ e−1 ∧ e−2 ∧ e−3 ∧ . . .

Now, we can define “field operators” Ψ and Ψ∗ acting on Ω as follows:

Ψ∗(ek)Ω = e0 ∧ e−1 ∧ e−2 ∧ · · · ∧ ek+1 ∧ek ∧ ek−1 ∧ . . . for k < 0

Ψ(ek′) Ω = ek′ ∧ e0 ∧ e−1 ∧ e−2 ∧ e−3 ∧ . . . for k ∈ Z(1.3.7)

We see that Ψ∗ acting on Ω creates positive energy states and Ψ annihilates negative energystates, i.e creates holes. In particular:

Ψ∗(g)Ω = 0, for g ∈ H−Ψ (f) Ω = 0, for f ∈ H+

Acting with these operators on Ω successively, we reach configurations of the Dirac sea i.e.infinite particle states, where finitely many positive energy states and almost all negativeenergy states are occupied. Formally, those are (linear combinations of) states of the form

Φ = ei0 ∧ ei1 ∧ ei2 ∧ . . . (1.3.8)

11


where (i0, i1, i2, ...) is a strictly decreasing sequence in Z with i(k+1) = ik− 1 for all sufficientlylarge indices k.

On the formal level, the transition from the hole theory to what we dubbed electron-positron picture now simply results in mapping (linear combinations of) states of the form(1.3.8) into the Fock-space F =

∧H+ ⊗

∧C(H−), such that the holes in the sea (∼ the

missing negative indices) are mapped to antiparticle-states and the positive energy states (∼positive indices) are mapped to the corresponding particle states. The idea is quite simple,although it’s somewhat tedious to write things down properly. We will spare the reader the for-mal details until Section 5.4. Anyways, after introducing the appropriate Fock-space structureon Dirac seas, it’s fairly easy to see that these assignments do indeed define an isomorphismand that under this isomorphism, the operator Ψ as defined in (1.3.7) acts just as the fieldoperator in (1.3.6).

So we note that the two descriptions are actually mathematically equivalent. If we carryout the whole process of second quantization in the electron-positron picture without any refer-ence to the Dirac sea whatsoever (and we will do that, in the next chapter), we’ll arrive at thevery same obstacles, in particular at the Shale-Stinespring condition for second quantizationof unitary operators. Personally, I would say that the meaning of these results remains moreobscure without reference to infinite particle states. Indeed, certain features of the theoryseem to indicate that the Dirac sea picture is a more honest description. For instance, in theformal quantization of the Dirac field, the vacuum-charge, just as the vacuum-energy above,appears as an infinite constant even after renaming creators and annihilators. Nevertheless,most physicists nowadays seem to strongly favor the electron-positron description, often callingthe idea of a Dirac sea an outdated concept that became obsolete once the theory was prop-erly understood. On the other hand, my personal teachers strongly advocate Dirac’s picture.What they ultimately have in mind is that the Dirac sea is just an approximate description ofa universe consisting of a very large, yet finite number of charged particles and that electronsor holes can be understood as deviations from an equilibrium state in which the particles areso homogeneously distributed that the net-interaction is zero and therefore the sea “invisible”.The Dirac sea formalism with infinitely many particles is then best understood as a thermody-namic limit of a fully interacting N-particle theory, whose consistent formulation has not yetbeen achieved. Until we have a complete and well-defined fully interacting theory of QuantumElectrodynamics, the choice between the two descriptions ultimately remains a matter of per-sonal taste. However, it should be clear that at this level of description no serious problemscan be solved by choosing one over the other.

12

Chapter 2

Polarization Classes and the RestrictedTransformation Groups

In this chapter, we give precise definitions for the concepts introduced in Chapter 1 and studythe restricted unitary- and general linear group of automorphism that do satisfy the Shale-Stinespring condition (1.2.10) and can be implemented on the fermionic Fock space.

By H we will always denote an infinite-dimensional, complex, separable Hilbert space.

Note: With the physical context in mind, we would think of H = H+ ⊕ H− as the spectraldecomposition w.r.to the free Dirac Hamiltonian and of the subspace H− as the projective“vacuum state”, corresponding to the Dirac sea. Unfortunately, the mathematical conventiondoesn’t follow the physical motivation and mathematicians for some reason prefer to work withthe subspace denoted by H+ instead of H−. We will follow this convention, in order to matchthe mathematical literature. Therefore, H+ and not H− will play the role of the “Dirac sea”in the following and most of the “mathematical” chapters.

2.1 Polarization ClassesDefinition 2.1.1 (Polarizations).

A polarization of H is an infinite-dimensional, closed subspace V ⊂ H with infinite-dimensionalorthogonal complement V ⊥.

Accordingly, we will call H a polarized Hilbert space if we have a distinct splittingH = V ⊕ V ⊥ for a polarization V .

The set of all polarizations of H is denoted by Pol(H).

By PV : H → H we denote the orthogonal projection of H onto V , so PV + PV ⊥ = 1H.

Definition 2.1.2 (Polarization Classes).On Pol(H), we introduce the equivalence relation

V ≈W ⇐⇒ PV − PW ∈ I2(H) (2.1.1)

The equivalence classes C ∈ Pol(H) are called polarization classes.

Lemma 2.1.3 (Characterization of ≈).For V,W ∈ Pol(H), the following are equivalent:

i) V ≈W

ii) PW⊥PV , PWPV ⊥ ∈ I2(H)

iii) PW |V→W is a Fredholm operator and PW⊥ |V→W⊥ ∈ I2(V ).

13

2.1. POLARIZATION CLASSES

Proof.

i)⇒ii): If V ≈W , i.e. PV − PW ∈ I2(H) then

PW⊥PV =(IdH − PW )PV = (PV − PW )PV ∈ I2(H)

PWPV ⊥ =PW (IdH − PV ) = −PW (PV − PW ) ∈ I2(H)

ii)⇒iii): We write the identity on H in matrix form as

IdH : V ⊕ V ⊥ →W ⊕W⊥ =

(PW |V→W PW |V ⊥→WPW⊥ |V→W⊥ PW⊥ |V ⊥→W⊥

)By ii), the off-diagonal terms are of Hilbert-Schmidt type, so the operator(

PW |V→W 00 PW⊥ |V ⊥→W⊥

)is a compact perturbation of the identity. Consequently, PW |V→W and PW⊥ |V ⊥→W⊥are Fredholm-operators.

ii)⇒i): If

PW⊥PV = (PV − PW )PV ∈ I2(H),

PWPV ⊥ = −PW (PV − PW ) ∈ I2(H)

then(PV − PW ) = (PV − PW )PV + PW (PV − PW ) ∈ I2(H)

iii)⇒ii): PWPV ⊥ ∈ I2(H) by assumption and as PW |V→W is a Fredholm operator, thecokernel of PWPV in W is finite-dimensional. Hence PW⊥PV is a finite-rank operator,in particular Hilbert-Schmidt.

By this Lemma, the following is well-defined:

Definition 2.1.4 (Relative Charge).For V,W ∈ Pol(H) with V ≈ W , we define the relative charge of V,W to be the Fredholmindex of PW |V→W . I.e.:

charge(V,W ) := ind(PW |V→W )

= dim ker(PW |V→W )− dim coker(PW |V→W )

= dim ker(PW |V→W )− dim ker((PW |V→W )∗

).

(2.1.2)

Lemma 2.1.5 (Properties of Relative Charge).The relative charge has the following intuitive properties:

i) charge(V,W ) = − charge(W,V )

ii) charge(V,W ) + charge(W,X) = charge(V,X) for V ≈W ≈ X ∈ Pol(H).

Proof. For V,W,X from the same polarization class in Pol(H)/ ≈ we find

charge(V,W ) + charge(W,X)

= ind(PW |V→W ) + ind(PX |W→X) = ind(PXPW |V→X)

= ind(PXPX + PX(PW − PX)

∣∣V→X

)= ind(PX |V→X)

= charge(V,X)

14

2.1. POLARIZATION CLASSES

The equality in the third line holds because PW −PX is of Hilbert-Schmidt type, in particulara compact perturbation, and therefore doesn’t change the index.As a special case we get

charge(V,W ) + charge(W,V ) = charge(V, V ) = 0,

which proves the first identity.

It follows from the Lemma that we get a finer equivalence relation ” ≈0 ”, by setting

V ≈0 W :⇐⇒ V ≈W and charge(V,W ) = 0. (2.1.3)

For our purposes, it is often more convenient to work with these equal charge classes. Also, thephysical principle of charge conservation tells us that the Dirac time evolution should preservethe relative charge.

In general, unitary transformations do NOT preserve the polarization class - they will mapone polarization class into another. This is the source of all evil when we try to implementthe unitary time evolution on Fock spaces. However, a unitary transformation induces at awell-defined map between polarization classes by U [V ] = [UV ] ∈ Pol(H)/ ≈. This is true,because PUV − PUW = UPV U

∗ − UPWU∗ = U(PV − PW )U∗ is of Hilbert-Schmidt type ifand only PV − PW is. The analogous map is also well-defined between equal charge classes asfollows, for example, from (2.2.8).

Definition 2.1.6 (Restricted Unitary Operators).

For polarization classes C,C ′ ∈ Pol(H)/ ≈ we define

Ures(H;C,C ′) = U : H → H unitary | ∀V ∈ C : UV ∈ C ′= U : H → H unitary | ∃V ∈ C with UV ∈ C ′

as the set of unitary operators mapping the polarization class C into C ′. If we want to restrictto equal charge classes, we write U0

res(H;C0, C′0) for C0, C

′0 ∈ Pol(H)/ ≈0 etc..

The definition can be immediately generalized to unitary maps between different Hilbert spacesH and H′.

Note that Ures(H;C,C ′) is not a group, unless C = C ′.However, they satisfy the composition property

Ures(H;C ′, C ′′)Ures(H;C,C ′) = Ures(H;C,C ′′). (2.1.4)

The group Ures(H;C,C) of unitary operators preserving a fixed polarization class C will beof crucial importance for our further discussion. In particular, as argued in the introductorysections, it will turn out that a unitary transformation can be implemented on the (standard)Fock space if and only if it’s compatible with the natural polarization H = H+ ⊕ H− in thesense that it preserves the polarization class C = [H+].

15

2.2. THE RESTRICTED UNITARY AND GENERAL LINEAR GROUP

2.2 The restricted Unitary and General Linear Group

We want to study the restricted unitary group consisting of those operators which do preservethe polarization class [H+] and will turn out to be implementable on the Fock space.

Definition 2.2.1 (Restricted Unitary Group).The group

Ures(H) := Ures(H; [H+], [H+])

is called the restricted unitary group on H.

We introduce the notation ε = P+ − P− for the sign of the free Dirac-Hamiltonian.

Lemma 2.2.2 (Characterization of Ures(H)).For a unitary operator U ∈ U(H) the following statements are equivalent:

i) U ∈ Ures(H)

ii) U+− ∈ I2(H−,H+) and U−+ ∈ I2(H+,H−)

iii) [ε, U ] ∈ I2(H)

Note that ii) is the infamous Shale-Stinespring condition, which appears very naturally in thissetting.

Proof. Let U ∈ U(H). Then:

U ∈ Ures(H) ⇐⇒ UH+ ≈ H+ ∈ Pol(H) ⇐⇒ P+ − UP+U∗ ∈ I2(H)

⇐⇒ P+U − UP+ ∈ I2(H)

⇐⇒ P+U(P+ + P−)− (P+ + P−)UP− = P+UP− − P−UP+ ∈ I2(H)

⇐⇒ U+− ∈ I2(H−,H+) and U−+ ∈ I2(H+,H−)

where we have used that PUV = UPV U∗ for any U ∈ U(H) and any subspace V ⊂ H.

Furthermore we compute

[ε, U ] = (P+ − P−)U − U(P+ − P−)

= (P+ − P−)U(P+ + P−)− (P+ + P−)U(P+ − P−)

= P+UP− − P−UP+ + P+UP− − P−UP+

= 2(P+UP− − P−UP+

).

And therefore1:1

4‖[ε, U ]‖22 =‖(P+UP− − P−UP+)∗(P+UP− − P−UP+)‖1

=‖(P−U∗P+ − P+U∗P−)(P+UP− − P−UP+)‖1

=‖P−U∗P+UP− + P+U∗P−UP+‖1

=‖U∗−+U+−‖1 + ‖U∗+−U−+‖1=‖U+−‖22 + ‖U−+‖22

(2.2.1)

as the traces of the odd parts vanish. This finishes the proof.

Thus, we can describe the restricted unitary group as

Ures(H) = U ∈ U(H) | [ε, U ] ∈ I2(H)= U ∈ U(H) | U+− and U−+ are Hilbert-Schmidt operators.

(2.2.2)

This alternative definition extends immediately to arbitrary isomorphisms. The unitary caseis the most relevant one for physical applications, but for the development of the mathematicalframework it is very natural and convenient to study the general case as well.

16


Definition 2.2.3 (Restricted General Linear Group).In analogy to (2.2.2) we define the group

GLres(H) := A ∈ GL(H) | [ε, A] ∈ I2(H)= A ∈ GL(H) | A+− and A−+ are Hilbert-Schmidt operators.

GLres(H) is called the restricted general linear group of H.

With respect to the decomposition H = H+ ⊕ H− we can write any A ∈ GL(H) in matrixform as

A =

(A++|H+→H+

A+−|H−→H+

A−+|H+→H− A−−|H−→H−

)=

(a bc d

). (2.2.3)

We call Aeven = A++ + A−− the even part and Aodd = A+− + A−+ the odd part of theoperator. Then, A is in GLres(H) if and only if Aodd ∈ I2(H), if and only if b and c areHilbert-Schmidt operators. In this case, as the odd part is just a compact perturbation, itfollows immediately that a and d are Fredholm operators with ind(a) = − ind(d), because

0 = ind(A) = ind

(a bc d

)= ind

(a 00 d

)= ind(a) + ind(d). (2.2.4)

N.B. that for A ∈ GL(H), A ∈ GLres(H) is not equivalent to [AH+] ≈ [H+] in Pol(H), butthe first implies the latter. Using Lemma, 2.1.3 iii), we see that for [AH+] ≈ [H+] it sufficesthat c is Hilbert-Schmidt and a a Fredholm operator which for general isomorphisms is lessrestrictive than A ∈ GLres(H).

2.2.1 Lie Group structure of GLres(H) and Ures(H)We recall a few basic definitions:

Definition 2.2.4. (Linear Banach Lie group)

Let (A, ‖·‖) a unital Banach algebra (cf. Def. 5.1.8) and A× its multiplicative group of units.

Recall that the exponential map exp : A → A×, x 7→∞∑k=0

xk

k! is well-defined and smooth with

D0 expx = x, ∀x ∈ A.

A linear Banach Lie group is a closed subgroup G of A× which is locally exponential. Thismeans that if we identify the Lie algebra of G with L(G) := x ∈ A | exp(x) ∈ G, theexponential map restricted to L(G) is a local diffeomorphism around 0.

Every (continuous) Lie group homomorphism ϕ : G1 → G2 between two linear Banach Liegroups is already smooth. There exists a unique Lie algebra homomorphism Lie(ϕ) with

expG2Lie(ϕ) = ϕ expG1

.

The last statement is proven in Appendix A.2.

Examples 2.2.5. (Linear Lie groups)

1. For every unital Banach algebra A, A× itself is a Lie group. It is always locally expo-nential due to the Inverse Mapping Theorem for Banach spaces (see for example [Lang]).

2. Let H a separable Hilbert-space. The set B(H) of bounded operators, together with theoperator norm, is a unital Banach algebra. The group U(H) of unitary operators is thena real linear Banach Lie group. Its Lie algebra can be easily identified as L(U(H)) =iX | X Hermitian operator on H, which is an algebra over R. The exponential mapis surjective onto U(H), as follows from the spectral theorem.

17


For our purposes, we consider the algebra Bε(H) of all bounded operators A : H → H with[ε, A] ∈ I2(H). We introduce a norm ‖·‖ε, defined by

‖A‖ε := ‖A‖+ ‖A+−‖2 + ‖A−+‖2 (2.2.5)

(this is equivalent to the norm ‖·‖+ ‖[ε, ·]‖2, also found in the mathematical literature). Thespace B(H) of bounded operators is complete in the operator-norm, and I2(H+,H−) as well asI2(H−,H+) are complete in the Hilbert-Schmidt norm. Thus, it is easily verified that Bε(H)equipped with the norm ‖·‖ε is a unital Banach algebra. Now, GLres(H) = B×ε is just themultiplicative group of units in Bε and Ures(H) is the closed subgroup of unitary operators inBε(H). It follows that GLres(H) is a complex linear Lie group and Ures(H) a real linear Liegroup. The corresponding Lie algebras are

gl1 =X bounded operator on H | [ε,X] ∈ I2(H) = B×ε (2.2.6)

ures =i · X Hermitian operator on H | [ε,X] ∈ I2(H). (2.2.7)

Geometrically, GLres(H) can also be understood as the complexification of Ures(H). In partic-ular, it inhibits Ures(H) as a real Lie subgroup and gl1

∼= ures ⊗R C.Finally, two more results about the topology of GLres(H).

Lemma 2.2.6 (Homotopy type of GLres).The map

A =

(a bc d

)7−→ a

from GLres(H) to the space Fred(H+) of Fredholm operators on H+ is a homotopy equivalence.In particular, GLres(H) has infinitely many connected components indexed by Z, correspondingto the index of the (++)-component a of A.

The last conclusion is true, because ind : Fred → Z is continuous. We denote the n-th con-nected component by GL(n)

res (H). In particular, GL0res(H) denotes the identity component of

GLres(H). For Ures(H) we use the analogous notations.

The Fredholm-index of a has a very important physical interpretation.For V ≈ H+ ∈ Pol(H) and A ∈ GLres(H), we find that

charge(AV,H+) = ind(P+|AV→H+) = ind(P+A|V→H+

)

= ind(P+A(P+ + P−)|V→H+

)= ind

((P+AP+ + P+AP−)|AV→H+

)= ind(P+AP+|V→H+

) = ind(P+AP+ P+|V→H+)

= ind(P+|V→H+) + ind(P+AP+|H+ → H+)

Thus:

charge(AV,H+) = charge(V,H+) + ind(a) (2.2.8)

So, physically, the index of the (++)-component corresponds to the net-charge that the trans-formation A “creates” from the vacuum (∼ H+, in this convention).

Lemma 2.2.7 (Homotopy Groups of GL0res).

The homotopy groups of the connected Lie group GL0res(H) are for k ≥ 0

π2k+1(GL0res) = 0 and π2k+2(GL0

res)∼= Z

In particular, GL0res(H) and U0

res(H) are simply-connected.

For the proofs of the Lemmatas see [PreSe] §6 or [Wurz06], for a more complete version.

18

2.3. THE RESTRICTED GRASSMANNIAN

2.3 The restricted Grassmannian

To the spectral decomposition H = H+ ⊕H− corresponds the polarization class[H+] ∈ Pol(H)/≈. This set of subspaces of H is known in the literature as the (restricted)Grassmannian of the Hilbert space and denoted by Gr(H) (the notations Grres(H) or Gr1(H)are also common). From the mathematical point of view, it’s a remarkably nice object. It canactually be given the structure of an infinite-dimensional complex (Kähler) manifold.

By Lemma 2.1.3 we can describe Gr(H) as the set of all closed subspaces W of H such that

i) the orthogonal projection P+ : W → H+ is a Fredholm operator

ii) the orthogonal projection P− : W → H− is a Hilbert-Schmidt operator.

This is the definition most commonly found in the mathematical literature (e.g. [PreSe], Def.7.1.1). Another way of saying the same thing is the following:A subspace W belongs to Gr(H) if and only if it’s the image of an operator w : H+ → H withP+ w ∈ Fred(H+) and P− w ∈ I2(H+,H−).

We can cover Gr(H) by the setsUW

W∈Gr(H)

, where

UW := W ′ ∈ Gr(H) | PW |W ′→W is an isomorphism . (2.3.1)

The elements of UW are “close” to W in the sense that they differ from W only by a Hilbert-Schmidt operator T : W →W⊥. We make this more precise:

Lemma 2.3.1 (Characterization of UW ).The set UW consists of all graphs of Hilbert-Schmidt operators W →W⊥. There is a bijectionbetween UW and I2(W,W⊥).

Proof. Let W ∈ Gr(H) a polarisation and T : W → W⊥ be a Hilbert-Schmidt operator. Letw : H+ →W an isomorphism with image W ⊂ H. Then:

Graph(T) = (w, T w) | w ∈W = im(w ⊕ T w∣∣W→W⊕W⊥) = im(w + T w

∣∣W→H).

(under the canonical identification W ⊕W⊥ ∼= H). Now, since W ∈ Gr(H), it follows thatP+(w + T w) = P+w + P+Tw is a Fredholm-operator and P−(w + T w) = P−w +P−Tw is Hilbert-Schmidt, hence Graph(T ) ∈ Gr(H). Furthermore, the orthogonal projectionPW∣∣Graph(T)→W is obviously an isomorphism, whose inverse is T , so that Graph(T ) ∈ UW .

Conversely, if W ′ ∈ UW , we can set

(PW |W ′→W )−1 = (IdW + T )|W→W ′ with T : W →W⊥.

Since W ′ ≈W , it follows from Lemma 2.1.3 that PW⊥ |W ′→W⊥ is Hilbert-Schmidt, hence

PW⊥ |W ′→W⊥ (IdW + T )|W→W ′= T

is Hilbert-Schmidt. Finally, it is easy to see that the assignments T ←→ W ′ = Graph(T ) areinverses of each other, so the statement of the Lemma is proven.

Proposition 2.3.2 (Manifold Structure of Gr(H)).Gr(H) is a complex Hilbert manifold modelled on I2(H+,H−).The sets UW ,W ∈ Gr(H) form an open covering of Gr(H).

Proof. [PreSe], Prop. 7.1.2

19

2.3. THE RESTRICTED GRASSMANNIAN

By definition, a unitary transformation maps Gr(H) into Gr(H) if and only if it is in Ures(H).Conversely, for any two polarizations W,W ′ of H there exists U ∈ U(H) with W ′ = UW .Then, if W,W ′ belong both to Gr(H), this transformation U is necessarily in Ures(H).

In other words: Ures(H), and consequently also GLres(H), act transitively on Gr(H). Thisleads us to a different description of the Grassmann manifold as a homogeneous space underUres(H) or GLres(H). The corresponding isotropy groups of H+ ∈ Gr(H) are

P :=

A =

(a bc d

)∈ GLres(H)

∣∣∣∣∣ c = 0

respectively

Q :=

U =

(a bc d

)∈ Ures(H)

∣∣∣∣∣ b = c = 0

This means:

GLres(H)/P ∼= Ures(H)/Q ∼= Gr(H) (2.3.2)

In particular, it follows that, just as GLres(H) and Ures(H), the Grassmannian Gr(H) has Zconnected components corresponding to the relative charges

charge(W,H+) = ind(P+|W→H+).

Alternatively, we could have noted that

charge(W ′,H+) = charge(W,H+), ∀W ′ ∈ UW ∀W ∈ Gr(H)

and hence the sets Gr(c)(H) := W ∈ Gr(H) | charge(W,H+) = c ∈ Z are open and disjoint.They are connected, because U0

res(H) acts continuously and transitively on each of them. Con-sequently, they correspond to different connected components.

It is a nice feature of the mathematical structure that different charges are separated topo-logically. This also reflects the physical intuition that a continuous time evolution preservesthe total charge i.e. that particles and anti-particles are always created in pairs.

Finally, we can also use the transitive action of Ures(H) on Gr(H) to define a Ures-invariant Hermitian form on the restricted Grassmannian. Since Gr(H) is a complex Hilbert-manifold modelled on I2(H+,H−), the tangent space at H+ ∈ Gr(H) is naturally isomorphicto I2(H+,H−). This space carries an inner product

(X,Y ) 7→ 2 tr(X∗Y ), (2.3.3)

which is obviously invariant under the action of the isotropy group U(H+) × U(H−). There-fore, we can extend (2.3.3) to a Ures-invariant Hermitian form on Gr(H) (by pull-backs withappropriate unitary transformations). In fact, this Hermitian form defines a Kähler structureon Gr(H), turning it into an infinite-dimensional Kähler-manifold ([PreSe] §7.8).

20

Chapter 3

Projective Representations and CentralExtensions

In this section we introduce the concept of central extensions of (Lie-)groups which is essentialfor the treatment of representations of Lie groups in a quantum mechanical setting. Again, wewill start with a rather intuitive approach to motivate the concept.

3.1 Motivation

We have argued – and will state rigorous results – that a unitary operator can be lifted to thefermionic Fock space F if and only it satisfies the Shale-Stinespring condition i.e. if and only ifit’s in Ures(H). In this case, we will see that the implementation, i.e. the “second quantization”of the operator is well-defined only up to a complex phase of modulus one. In this chapter, weare concerned with this phase-freedom. Suppose we are trying to tackle this latter problem.Suppose we choose any prescription for fixing the phase and denote by Γ(U) the correspondinglift of U ∈ Ures to an operator on the Fock space F . This yields a map

Γ : Ures(H)→ U(F) (3.1.1)

from Ures(H) into the group of unitary automorphisms of the Fock space. Such a map is what’susually - and somewhat mysteriously - called a “second quantization” of the unitary operators.However, fixing the phases of the lifts in some arbitrary way, we have no reason to expect thatthese lifts will preserve the group structure: for any U, V ∈ Ures(H), Γ(U)Γ(V ) and Γ(UV ) areboth implementations of the same unitary operator but will, in general, differ by a complexphase. In other words, Γ will fail to be a representation of the restricted unitary group Ures

on F . Instead, we merely get a projective representation of Ures on the projective Fock spaceP(F) = F \ 0 mod C.

Now, we would like to ask: is it possible to fix the phases of the lifts in such a way as topreserve the group structure? In other words:

Does a proper representation of Ures(H) on the Fock space exist?

The answer to this question will require a more sophisticated analysis. First, we observe thatwe can always write

Γ(U)Γ(V ) = χ(U, V ) Γ(UV ) (3.1.2)

for U, V ∈ Ures with χ(U, V ) ∈ U(1). This defines a map χ : Ures ×Ures → U(1).Reasonably, we demand Γ(1) = 1F , which implies χ(1,1) = 1. Such a map is called a 2-cocycleor a factor set. Using this factor set, we can define a group

U(1)×χ Ures

21

3.2. CENTRAL EXTENSIONS OF GROUPS

as the direct product U(1)×Ures with the multiplication given by

(a, U) ·χ (b, V ) :=(χ(U, V ) ab, UV

)(3.1.3)

and set Γ((a, U)

):= aΓ(U). But the multiplication on U(1) ×χ Ures was defined precisely

in such a way as to compensate the cocycle coming from Γ and make Γ a homomorphism ofgroups:

Γ((a, U)(b, V )

)= Γ

((χ(U, V ) ab, UV )

)= ab χ(U, V )Γ

(UV

)= abΓ(U) Γ(V ) = Γ

((a, U)

)Γ((b, V )

)The group U(1)×χ Ures is called a central extension of Ures by U(1).

This is quite nice. Indeed, there is an intimate connection between central extensions and theproblem of lifting projective representations to proper representations that we are going toexploit in the following chapter. Of course, things aren’t quite as simple as we have presentedthem so far and thus a more general approach is required.

3.2 Central Extensions of GroupsThroughout this section, let G be an arbitrary group and A an abelian group. The trivialgroup, consisting of the neutral element only, is denoted by 1. All definitions and results applydirectly to topological groups and Lie groups, if the appropriate structures on the groups andcontinuity, respectively smoothness, of the maps are implied.

Definition 3.2.1 (Central Extension).A central extension of G by A is a short exact sequence of group homomorphisms

1 −→ Aı−−→ E

π−−→ G −→ 1

such that ı(A) is in the center of E (i.e commutes with all the elements in E).

Let’s dissect this abstract definition: The sequence being exact means that the kernel ofeach map equals the image of the previous map. So ı must be injective, π surjective andker(π) = im(ı) ∼= A. Thus, E is a covering of G and the preimage of every g ∈ G is isomor-phic to A. The requirement that ı(A) is central in E is crucial in the context of studying(projective) representations of G, since then, Schurr’s Lemma ensures that in any irreduciblerepresentation, the images of A in E will be constant multiples of the identity. In a certainsense, A will carry the information about the phases that lead to ambiguities when trying tolift a projective representation of G to a proper one.

Examples 3.2.2 (Trivial extension and universal covering group).

1. A trivial extension has the form

1 −→ Aı−−→ A×G pr2−−−→ G −→ 1

where E is just the direct product of A and G, i(a) = (a, 1), ∀a ∈ A and pr2 is theprojection onto the second entry.

2. Let G be a connected topological group, G the universal covering group andA = π1(G) ∼= Cov(G,G) the fundamental group of G. Then

1 −→ π1(G)ı−−→ G

π−−→ G −→ 1

is a central extension of G by π1(G), where π is the covering homomorphism and ı isgiven by the action of the covering group Cov(G,G) ∼= π1(G) on 1 ∈ G.

22

3.2. CENTRAL EXTENSIONS OF GROUPS

Proof. It is easily checked that the sequence is exact. Furthermore, the orbit of 1Gunder the action of Cov(G,G) is by definition the preimage of 1G under π. Thus:i(π1(G)) = π−1(1G) = ker(π). The interesting part is to show that ı(π1(G)) is central inG. To this end, note that ı(π1(G)) = π−1(1G) is a discrete set by definition of coveringspaces. As the kernel of π, it is also a normal subgroup of G. Now, for any fixeda ∈ ı(π1(G)) we can consider the map

G 3 g 7→ g−1ag.

This is a continuous map from G into ı(π1(G)), sending 1 to a. As G is connected andı(π1(G)) is discrete, the map must be constant. Thus, g−1ag = a ∀g ∈ G, i.e. a belongsto the center of G.

3. As a special case of 2. , we can consider the well known covering

1 −→ ±1 → SU(2)→ SO(3) −→ 1

It is a well-known fact (in physics prominently drom the discussion of spin in non-relativistic quantum mechanics) that there is no irreducible, 2-dimensional unitary rep-resentation of SO(3). However, there exists one of its universal covering group SU(2)(generated by the Pauli matrices).

Definition 3.2.3 (Equivalence of Central Extensions).

Two central extension E and E′ of a group G by A are equivalent, if there exists an isomorphismϕ : E → E′ which is compatible with the extensions i.e. such that the following diagramcommutes:

1 // A //

Id

E

ϕ

// G

Id

// 1

1 // A // E′ // G // 1

Lemma 3.2.4 (Trivial Extensions).

A central extension 1 −→ Aı−−→ E

π−−→ G −→ 1 is equivalent to the trivial extension if andonly if there is a homomorphism σ : G→ E with π σ = IdG.

In other words: σ is a section of G in E which is also a homomorphism of groups. In thiscase, σ is called a splitting map and the extension is said to split.

If G and E are Lie groups, a section σ is called a splitting map if it is also a Lie grouphomomorphism.

Proof. If the extension is trivial i.e. E ∼= A×G, set σ(g) := (1, g) ∈ A×G.Conversely, suppose there exists σ : G→ E as above. Set ϕ : A×G→ E, (a, g) 7→ ı(a)σ(g).It is easily checked that this is a homomorphism compatible with the extension in the sense ofDef. 3.2.3. Furthermore, ϕ is bijective, since for every g ∈ G and ξ ∈ π−1(g) ⊂ E there is oneand only one a ∈ A with ξ = ı(a)σ(g).

23

3.3. PROJECTIVE REPRESENTATIONS

3.3 Projective RepresentationsDefinition 3.3.1 (Projective Hilbert space).Let H be a complex Hilbert space of finite or infinite dimension.The projective Hilbert space P(H) is the space of rays in H, i.e.

P(H) := (H\0)/C×

where the equivalence relation is given by ϕ ∼ λϕ , for λ ∈ C\0. The topology on P(H) isthe quotient topology, induced by the quotient map γ : H → P(H).We will also write ϕ instead of γ(ϕ).

Obviously, the projective Hilbert space is not a linear space. However, transition probabilitiesare still well-defined. By this, we mean the map δ : P(H)× P(H)→ [0, 1] defined by

δ(ϕ, ψ) :=|〈ϕ,ψ〉|2

‖ϕ‖2‖ψ‖2(3.3.1)

where 〈·, ·〉 is the Hermitian scalar product on H.

Definition 3.3.2 (Projective Automorphisms).A projective automorphism is a bijective map T : P(H) → P(H) preserving the transitionprobability i.e. satisfying

δ(T ϕ, T ψ) = δ(ϕ, ψ) , ∀ϕ, ψ ∈ P(H).

We denote the set of all projective automorphisms by Aut(P(H)).

If U is a unitary map on H, we can define U on P(H) by

U(γ(ϕ)) = γ(U(ϕ)

). (3.3.2)

Clearly, this is a projective automorphism. Thus, we get a group-homomorphism

γ : U(H)→ Aut(P(H)) ,

U 7→ γ(U) = U .

As the projective Hilbert space doesn’t care for multiplicative constants, the same definitionalso works for anti-unitary maps U on H, i.e. anti-linear maps satisfying 〈Ux,Uy〉 = 〈x, y〉for x, y ∈ H. As a matter of fact, every projective automorphism comes from a unitary oranti-unitary map on H:

Theorem 3.3.3 (Wigner, 31).For every projective automorphism T ∈ Aut(P(H)) there exists a unitary or an anti-unitaryoperator U on H with T = γ(U).

The projective automorphisms coming from unitary transformations on H form a subgroupof Aut(P(H)), denoted by U(P(H)). We can express this reasoning in the language of centralextensions:

Lemma 3.3.4 (Unitary Projective Automorphisms).The sequence

1 −→ U(1) −→ U(H)γ−−→ U(P(H)) −→ 1 (3.3.3)

defines a central extension of U(P(H)) by U(1) which is non-trivial.

Proof. The only part in proving that the sequence is exact and central that might be non-trivial is to identify ker(γ) with U(1) · Id ⊂ U(H). "⊇" is clear. For "⊆" pick U ∈ ker(γ).Then for any ϕ ∈ H : γ(U)(γ(ϕ)) = γ(Uϕ) = γ(ϕ)⇒ ∃λ ∈ C : Uϕ = λϕ.Since U is unitary, |λ| = 1 i.e. λ ∈ U(1). If we take any other ψ ∈ H which is not a multiple

24


of ϕ it is by the previous consideration also an eigenvector to some eigenvalue λ′ ∈ U(1); butso is the sum ϕ+ψ (with eigenvalue µ). It follows that U(ϕ+ψ) = µ(ϕ+ψ) = λϕ+ λ′ψ andthus, by linear independence, λ′ = λ = µ. Hence, U = λId.

To prove that the central extension is not trivial, we embed C2 in H be fixing any 2-dimensional subspace V ⊂ H. Now consider the subgroup of all unitary operators on Hleaving V invariant, i.e. U ∈ U(H) | U(V ) = V =: V. On V, we introduce an equivalencerelation and identify two operators if they agree on V . Then V/ ∼ ∼= U(V ) ∼= U(2).Now all the homomorphisms descend to a central extension

1 −→ U(1) −→ U(2) −→ U(P(C2)) −→ 1.

Note that P(C2) = CP1 ∼= S1, so its unitary group is just the isometry group of the sphere, i.e.SO(3). Also, U(1) × SU(2) ∼= U(2) by (eiφ, U) → eiφ/2 U , for φ ∈ [0, 2π). Thus, if the centralextension (3.3.3) was trivial with splitting map σ, this σ would descend to a spitting map for

1 −→ U(1) −→ U(1)× SU(2) −→ SO(3) −→ 1.

But then, the second component of the homomorphism σ : SO(3)→ U(1)×SU(2) is a splittingmap for the universal covering

1 −→ ±1 −→ SU(2) −→ SO(3) −→ 1

We know, however, that such a splitting map cannot exist, because the covering of SO(3) bySU(2) is not trivial. Thus, we get a contradiction.

Lifting projective representations

Now we are able to formulate the problem of lifting projective representations in a moreprecise way: Given a projective representation Γ : G → U(P(H)), is there a representationρ : G→ U(H) such that γ ρ = Γ, i.e. such that the following diagram commutes?

G

Γ

ρ

yy1 // U(1) // U(H)

γ // U(P(H)) // 1

In general, the answer is NO. However, as we have suggested in the introducing remarks, therealways exists a central extension G of G such that the projective representation of G lifts to aproper representation of G.

Lemma 3.3.5 (Lifting Projective Representations).Let G be a group and Γ : G→ U(P(H)) a homomorphism. There exists a central extension Gof G by U(1) and a homomorphism Γ : G→ U(H), such that the following diagram commutes:

1 // U(1)ı //

Id

G

Γ

π // G

Γ

// 1

1 // U(1) // U(H)γ // U(P(H)) // 1

(3.3.4)

Γ is a unitary representation of G on H. The group G is called the deprojectivization of G.

25


Proof. We defineG := (U, g) ∈ U(H)×G | γ(U) = Γ(g).

This is a subgroup of U(H)×G. The inclusion U(1) 3 λ ı7→ (λ · Id, 1) and the projection ontothe second component π = pr2 : G → G are homomorphisms that make the upper row of thediagram (3.3.5) into a central extension. The projection onto the first component defines arepresentation Γ := pr1 : G→ U(H) which, by construction, satisfies γ Γ = Γ π.

So far, this is a pure algebraic statement. In the more interesting cases, where G isa topological group or even a Lie group, we will have to check if and how the results arecompatible with the topological structure, respectively the Lie group structure, of G and itscentral extension. Here, we summarize the main results. A more complete treatment can befound in [Scho], for example.

1) If G is a topological group, G can be given the structure of a topological group as a subgroupof U(H)×G. Then, if Γ is continuous, so is Γ.

2) If G is a finite-dimensional Lie group, then G can be given the structure of a Lie group sothat the upper sequence in (3.3.4) becomes a sequence of differentiable homomorphisms. IfΓ is smooth (in a strong sense), so is Γ.1

3) If we have to deal with infinite-dimensional Lie groups – and we do – the Lie group structureof the central extension is not generally for free. Fortunately, things work out nicely in thecases relevant to our discussion.

In conclusion, if the central extension G carries the structure of a Lie group with a repre-sentation Γ on H, any prescription for lifting the projective action of G to the Hilbert-spacecorresponds to a section σ : G→ G, via

ρ := Γ σ : G→ U(H). (3.3.5)

Conversely, every attempt to lift the action to the Hilbert space, i.e. every (continuous) mapρ : G → U(H) defines a section in G, by assigning to g ∈ G the unique element g in π−1(g)

with Γ(g) = ρ(g). Now, ρ := Γ σ defines a (continuous/smooth) representation of G on H ifand only if the section σ is a (continuous/smooth) homomorphism of groups i.e. if and only ifthe central extension is trivial as an extension of groups/topological groups/ Lie groups. Wesummarize this insight in the following proposition.

Proposition 3.3.6 (Lifting Projective Representations).

Let G be a topological group. A projective representation Γ : G → U(P(H)) can be lifted to a(continuous) unitary representation ρ : G → U(H) with γ ρ = Γ if and only if the centralextension 1 −→ U(1) −→ G −→ G −→ 1 splits by a (continuous) section i.e. is trivial.

Proof. By Lemma (3.2.4), the central extension is (algebraically) trivial if and only if thereis a section σ : G → G which is also a homomorphism of groups. Then ρ := Γ σ is ahomomorphism with

γ ρ = γ Γ σ = Γ π σ = Γ.

If the section σ is continuous, so is ρ.Conversely, if ρ is a unitary representation of G, then σ(g) := (ρ(g), g) ∈ G is a section in Gand a homomorphism of groups and thus the desired splitting map.

1This statement seems rather harmless, but it’s quite the opposite. Indeed, it requires the solutionof one of the famous “Hilbert problems”: every topological group which is also a finite-dimensionaltopological manifold is already a Lie group. This theorem was proven by Montgomory and Zippin in1955.

26

3.4. COCYCLES AND COHOMOLOGY GROUP

3.4 Cocycles and Cohomology GroupIn the previous section, we have seen that lifts of a projective representation of a (Lie) group Gto an action on the Hilbert-space H (not necessarily a representation) correspond to sectionsin a central extension of G. We want to study this more thoroughly.

Let1 −→ A

ı−−→ Eπ−−→ G −→ 1

be a central extension of G by A. Let τ : G→ E be a map with

π τ = IdG and τ(1) = 1. (3.4.1)

The map might be defined in a neighborhood of the identity, only.τ will in general fail to be a homomorphism. Nevertheless:

π(τ(g)τ(h)) = π(τ(gh)) = gh, ∀g, h ∈ G.

Therefore, there exists χ(g, h) ∈ A with

τ(g)τ(h) = χ(g, h)τ(gh). (3.4.2)

This defines a map χ : G×G→ A. Obviously, this χ satisfies

χ(1, 1) = 1. (3.4.3)

Furthermore: τ(x)τ(y)τ(z) = χ(x, y)τ(xyb)τ(z) = χ(x, y)χ(xy, z)τ(xyz) and similarly: τ(x)τ(y)τ(z) =

τ(x)χ(y, z)τ(yz) = χ(x, yz)χ(y, z)τ(xyz), from which we deduce:

χ(x, y)χ(xy, z) = χ(x, yz)χ(y, z), ∀ x, y, z ∈ G. (3.4.4)

Now It’s possible that we’ve just made a “bad” choice for τ and that there really does exista (local) section τ ′ which is also a homomorphism, i.e. for which the corresponding cocyclevanishes. Let’s fix this by writing τ ′(x) = τ(x)λ(x) with a suitable function λ : G→ A.Then: τ ′(x)τ ′(y) = τ(x)τ(y)λ(x)λ(y) = τ(xy)χ(x, y)λ(x)λ(y). But also, since τ ′ is ahomomorphism: τ ′(x)τ ′(y) = τ ′(xy) = τ(xy)λ(xy). Together, this implies for λ:

λ(xy) = χ(x, y)λ(x)λ(y), ∀x, y, z ∈ G. (3.4.5)

This motivates the following definition:

Definition 3.4.1 (Cocycles and Second Cohomology Group).

i) A map χ : G × G → A satisfying (3.4.3) and (3.4.4) is called a factor set or a 2-cocycleon G with values in A.

ii) A 2-cocycle χ : G × G → is called trivial if there exists a map λ : G → A such thatλ(xy) = χ(x, y)λ(x)λ(y) ,∀x, y, z ∈ G.

iii) Furthermore, we define the second cohomology group of G with coefficients in A as the setof all 2-cocycles on G with values in A modulo trivial cocycles, i.e.

H2(G,A) := χ : G×G→ A | χ is a cocycle/ ∼ (3.4.6)

where χ1 ∼ χ2 :⇐⇒ χ1χ−12 is trivial.

27

3.4. COCYCLES AND COHOMOLOGY GROUP

We have seen that given a central extension 1 −→ Aı−−→ E

π−−→ G −→ 1, every sectionτ : G → E with π τ = IdG and τ(1) = 1 defines a cocycle χ : G×G → A. Different choicesfor τ lead to equivalent cocycles. Conversely, given a cocycle χ, we can define a group A×χ Gas the direct product A×G with the multiplication

(a, x)(b, y) :=(χ(x, y)ab, xy

). (3.4.7)

This is a central extension of G by A with the cocycle χ coming from the obvious sectionτ(x) := (1, x) (c.f. the construction in the introduction of this chapter).If the cocycle χ comes from a (global) section τ : G→ E as above, the central extensions

1 −→ Aı−−→ E

π−−→ G −→ 1

and1 −→ A −→ A×χ G

pr2−−−→ G −→ 1

are equivalent by the isomorphism ϕ : A ×χ G → E, ϕ((a, c)

):= ı(a) · τ(x). The group-

multiplication in A×χ G is defined just in such a way as to cancel the cocycle coming from τ

on the right-hand-side. We reserve for the reader the little joy of checking for him- or herself howeverything is designed to work out nicely. Now it’s an easy exercise to show that for cocyclesχ1 and χ2, the corresponding groups A ×χ1

G and A ×χ2G are equivalent (isomorphic) as

central extensions if and only if χ1 and χ2 are equivalent as cocycles. Altogether, this yieldsthe following theorem:

Theorem 3.4.2 (Central Extensions correspond to Cohomology Classes).There is a one-to-one correspondence between equivalence classes of central extension of G byA and the second cohomology classes of G with values in A.

In particular, a central extension is trivial, if and only if it allows a (global) section whosecorresponding cocycle is trivial. In our context, this means that a 2-cocycle or, more generally,the corresponding cohomology group represents the algebraic or, if questions of continuity areinvolved, topological obstructions to lifting projective representations to proper representationson the Hilbert space.

Note that the theorem is so far a pure algebraic statement! If we are dealing with Liegroups and have to take topological aspects into account, things aren’t quite as easy and thearguments above will, in general, work only locally. In fact, there might not be any continuoussection τ : G→ E with π τ = IdG and τ(1) = 1. Therefore, it is usually more convenient todiscuss cocycles of the corresponding Lie algebras, which may be thought of as the infinitesimalversion of the Lie group cocylces and are already determined by a local section τ defined insome arbitrarily small neighborhood of the identity of the Lie group G. We will do this in thenext chapter. However, the following is true:

Proposition 3.4.3 (Lie Group Extensions and local Cocycles).Let 1 → A → E → G → 1 be a central extension of a connected (finite- oder infinite-dimensional) Lie group G by the abelian Lie group A. Then, E carries the structure of aLie group such that the central extension is smooth if and only if the central extension can bedescribed by a cocycle χ : G×G→ A which is smooth in a neighborhood of (e, e) ∈ G×G.For Banach Lie groups, the statement applies also to non-connected G.

For the proof we refer to [Ne02] Prop. 4.2 and [TW87] Prop. 3.11 .

28

3.5. CENTRAL EXTENSIONS OF LIE ALGEBRAS

3.5 Central Extensions of Lie Algebras

In the following, we assume that G is a locally exponential Lie group (finite- oder infinite-dimensional) and that the assignment G→ Lie(G) of a Lie group to its Lie algebra is functorialin the following sense: If G1, G2 are Lie groups and ϕ : G1 → G2 a Lie group homomorphism,then their exists a unique Lie algebra homomorphism Lie(ϕ) = ϕ such that the followingdiagram commutes:

G1ϕ // G2

Lie(G1)

exp

OO

Lie(ϕ) // Lie(G2)

exp

OO (3.5.1)

Usually, we’d like to define a Lie group homomorphism as a continuous homomorphism betweenLie groups. However, for some “exotic” examples, it can be necessary to explicitly demanddifferentiability of the homomorphism at the identity, in order to get the functorial propertydefined above.

Definition 3.5.1 (Central Extension of Lie Algebras).Let a be an abelian Lie algebra and g a Lie algebra over R or C (the dimensions may beinfinite). A central extension of g by a is an exact sequence of Lie algebra homomorphisms

0 −→ ai−→ h

π−→ g −→ 0

s.t. i(a) ⊂ h is central in h, i.e. if [i(X), Y ] = 0 for all X ∈ a and Y ∈ h.

Proposition 3.5.2 (From Lie Groups to Lie Algebras).Let A, E and G finite-dimensional Lie groups and

1 −→ Ai−→ E

π−→ G −→ 1

a central extension of Lie groups. Then

0 −→ Lie(A)ı−→ Lie(E)

π−→ Lie(G) −→ 0 (3.5.2)

is a central extension of the corresponding Lie algebras.

Now, for a central extension of a Lie algebras

0 −→ ai−→ h

π−→ g −→ 0

we can always define a linear map β : g→ h satisfying π β = Id.Analogously to the central extensions of groups, the central extension of Lie algebras is equiv-alent to the trivial extension h = g ⊕ a if and only if β can be chosen to be a Lie algebrahomomorphism, i.e. such that [β(X), β(Y )] = β([X,Y ]), ∀X,Y ∈ g. In general, how β failsto be a Lie algebra homomorphism is expressed by the skew-symmetric map

c(X,Y ) := [β(X), β(Y )]− β([X,Y ]). (3.5.3)

29


c : g× g→ a has the following properties:

i) c is bilinear and skew-symmetric

ii) c (X, [Y,Z]) + c (Y, [Z,X]) + c (Z, [X,Y ]) = 0.(3.5.4)

In particular, if the central extension of Lie algebras comes from a central extension of Liegroups, any differentiable local section τ : G → E as in (3.4.1) defines a corresponding Liealgebra section by β := Deτ (under identification of the Lie algebras with the tangent space ofthe corresponding Lie groups at the identity e). But as τ fails to be a Lie group homomorphism,Dτ will fail to be a homomorphism of the Lie algebras.

Definition 3.5.3 (Lie Algebra Cocycles).Let a, g be two Lie algebras, a abelian. A map c : g× g→ a satisfying the conditions (3.5.4)is called a Lie algebra 2-cocycle or simply a cocycle.

Proposition 3.5.4 (Computation of Lie Algebra Cocycles).If χ is a (local) Lie group cocycle coming from the (local) section τ , then the corresponding Liealgebra cocycle coming from τ can be computed from χ as

c(X,Y ) =∂

∂t

∂

∂s

∣∣∣t=s=0

χ(esX , etY ) − ∂

∂t

∂

∂s

∣∣∣t=s=0

χ(etY , esX) (3.5.5)

Proof. : Here, we write τ for the Lie algebra map, corresponding to Deτ . We compute:

∂

∂t

∂

∂s

∣∣∣t=s=0

χ(esX , etY ) − ∂

∂t

∂

∂s

∣∣∣t=s=0

χ(etY , esX)

=∂

∂t

∂

∂s

∣∣∣t=s=0

τ(esX)τ(etY )τ(esXetY )−1 −∣∣∣t=s=0

τ(etY )τ(esX)τ(etY esX)−1

= τ(X)τ(Y ) − τ(XY ) − τ(Y )τ(X) + τ(Y X)

= τ(X)τ(Y )− τ(Y )τ(X)− ( τ(XY )− τ(Y X) )

= [τ(X), τ(Y )]− τ([X,Y ]) = c(X,Y ).

Just as for groups, there is a correspondence between central extensions of Lie algebras andcocycles which is 1-to-1 modulo trivial extensions/cocycles. Given bilinear form b c : g×g→ a,consider the vector space h := g⊕ a and define a bilinear map [·, ·]c on h by

[X1 ⊕ Y1, X2 ⊕ Y2]c := [X1, X2]g + c(X1, X2) (3.5.6)

for X1, X2 ∈ g, Y1, Y2 ∈ a. It is straight forward to check that [·, ·]c is a Lie bracket on h, ifand only if c is a cocycle. In this case, h becomes a Lie algebra and projection onto the firstcomponent makes

0 −→ a −→ hpr1−→ g −→ 0

a central extension of g by a. Conversely, if c comes from a central extension and a linear mapβ : g→ h, the algebra h is isomorphic to g⊕ a as a linear space, by the isomorphism

F : g× a→ h, X ⊕ Y = (X,Y ) 7→ β(X) + Y .

30


And equipped with the Lie bracket (3.5.6), g⊕ a becomes a Lie algebra and F an isomorphismof Lie algebras:

[F(X1, Y1),F(X2, Y2)] = [β(X1) + Y1, β(X2) + Y2] = [β(X1), β(X2)]

= β([X1, X2]) + c(X1, X2) = F([X1, X2]) + c(X1, X2))

= F ([(X1, Y1), (X2, Y2)]c).

We summarize the results in the following lemma:

Lemma 3.5.5 (Central Extensions and Lie Algebra Cocycles).Every central extension of Lie algebras comes from a cocycle.Conversely, every cocycle c : g× g→ a induces a central extension of g by a as above.

Note that in all of these construction a particular choice of β was involved. What if we choosea different linear map β′ : g→ h with π β′ = Id? Then, since π (β′− β) = 0, the differenceof β and β′ is a linear map with values in a (identified with the corresponding subalgebra inh), i.e.

β′ − β = µ : g→ a ∼= i(a) ⊂ h

For the corresponding cocycles, this means

c′(X,Y ) : = [β′(X), β′(Y )]− β′([X,Y ])

= [β(X) + µ(Y ), β(Y ) + µ(Y )]− β([X,Y ])− µ([X,Y ])

= [β(X), β(Y )]− β([X,Y ])− µ([X,Y ])

= c(X,Y )− µ([X,Y ])

since the image of µ is central in h. We deduce:

Theorem 3.5.6 (Triviality of Lie Algebra Extensions).Let c be a Lie algebra cocycle for the central extension h of g. The following are equivalent:

i) There exists a section β : g→ h that is also a Lie algebra homomorphism, i.e. a splitting-map for the central extension.

ii) The cocycle c defined by β vanishes.

iii) There exists a linear map µ : g→ a with c(X,Y ) ≡ µ([X,Y ]).

iv) The central extension is trivial, i.e. h ∼= g⊕ a as Lie algebras.

In particular, if τ : G → H is a splitting map for a central extension of Lie groups, β := τ isa splitting map for the corresponding central extension of Lie algebras.

These considerations lead us straight to the mathematical concept of cohomology groups.

Definition 3.5.7 (Second Cohomology Group).Let g be a Lie algebra and a an abelian Lie algebra.

Let Z2(g, a) be the set of all 2-cocycles on g with values in a.

Let B2(g, a) = c ∈ Z2(g, a) | ∃µ ∈ Hom(g, a) : c(X,Y ) = µ([X,Y ])

Let H2(g, a) := Z2(g, a)/B2(g, a).

H2(g, a) is the second cohomology group of g with values in a.

31


Theorem 3.5.8 (Central Extensions correspond to Cohomology Classes).The cohomology group H2(g, a) is in one-to-one correspondence with the set of equivalenceclasses of central extensions of g by a.

Let’s summarize the insights we’ve got so far. We started with a Lie group G and aprojective unitary representation Γ of G on a projective Hilbert space P(H). In general,we cannot expect to be able to lift the projective representation to a proper representationon H. However, there exists a central extension G of G by U(1) that does have a unitaryrepresentation Γ on the Hilbert space, with γ Γ = Γ 3.3.5. Conversely, any prescription forlifting the projective representation Γ to U(H), i.e. any choice of phases for the lift, correspondsto a section τ of G in G. In particular, for any such section, Γ τ defines a lift as desired.This will not be a representation, though, unless τ is a continuous homomorphism. The wayin which this fails to be a representation, or equivalently, in which the section fails to be ahomomorphism, is encoded in the corresponding cocycle. We have deduced that the followingstatements are equivalent:

1. There exists unitary representation ρ : G→ U(H) with γ ρ = Γ.

2. There exists a continuous section σ : G→ G which is also a homomorphism [Cor.3.3.6].

3. The central extension 1 −→ U(1) → G → G → 1 is trivial i.e. splits by a continuoussection [Lem. 3.2.4 ].

4. Any Lie group cocycle corresponding to the central extension above is trivial i.e. corre-sponds to 0 in H2(G,U(1)) [Thm. 3.5.8 ].

The “infinitesimal version” of this, so to speak, is the corresponding central extension of Liealgebras (3.5.2) and its Lie algebra cocycle (3.5.3). We have explained why it is more convenientto work with Lie algebras in general, but we still have to clarify how exactly this infinitesimalversion is related to our original problem. So far it is clear that if σ : G→ E is a section of Gin E which is also a Lie group homomorphism (i.e. a splitting map for the central extensionof Lie groups), then σ = Lie(σ) is a continuous Lie algebra homomorphism with π β = Id

and thus a splitting map, defining a trivialization for the corresponding central extension ofLie algebras. Therefore, triviality of the Lie algebra extensions is a necessary condition for thetriviality of the Lie group extension. The converse is not true in general, but at least underrather mild assumptions:

Proposition 3.5.9 (Lie Group Homomorphisms from Lie Algebra Homomorphisms).Let G1, G2 two Lie groups with Lie algebra g1 and g2, respectively. Assume that G1 is connectedand simply connected, and G2 is regular. Then, for every continuous Lie algebra homomor-phism F : g1 → g2 there exists a unique Lie algebra homomorphism f : G1 → G2 with F = f .

Proof. This is theorem 8.1 in [Miln]

From this, we conclude:

Proposition 3.5.10 (Triviality of Extensions).Let 1 −→ A

ı−−→ Eπ−−→ G −→ 1 be a central extension of G by A. Let G and E satisfy the

requirements of the previous proposition. Then, the central extension splits by a continuoussection if and only if the corresponding central extension of Lie algebras (3.5.2) splits by acontinuous Lie algebra homomorphism.

32


Proof. If the central extension of Lie groups splits by a section σ : G→ E, then β = σ is a Liealgebra homomorphism with π β = Id and thus a splitting map for the LIe algebra extensionwhich trivializes the central extension. Conversely, let β : Lie(G) → Lie(E) be a Lie algebrahomomorphism with π β = Id. Using the previous proposition, we can deduce that thereexists a Lie group homomorphism σ : G → E with β = σ. Since ˙(π σ) = π β = Idg = ˙IdG,the uniqueness part of the proposition yields π σ = IdG and thus σ is a splitting map for thecentral extension of Lie groups.

For completeness, we state Bargmanns theorem which is probably the main result of the theoryof central extensions, and give a brief sketch of the proof. We will not make any further use ofthis result, however, it’s certainly of great interest on its own. For the complete, original proofwe refer the reader to [Bar54].

Theorem 3.5.11 (Bargmann, 1954).Let G be a connected and simply connected finite-dimensional Lie group with

H2(Lie(G),R) = 0

Then, every projective unitary representation of G on P(H) can be lifted to a unitary represen-tation on H.

Proof. By Lemma(3.3.5) there exists a central extension G ofG by U(1), such that the followingdiagram commutes:

1 // U(1)i //

Id

G

Γ

π // G

σ

Γ

// 1

1 // U(1) // U(H)γ // U(P(H)) // 1

Now, one has to show that G can be given the structure of a (dim(G) + 1)-dimensional Liegroup. Thus, to this central extension of Lie groups corresponds a central extension of Liealgebras. Since H2(Lie(G),R) = 0, this central extension splits. By the previous Lemma, thisimplies that the central extension of Lie groups split, i.e. there is a differentiable homomorphismσ : G→ G with π σ = IdG. Then, Γ σ is the postulated lift.

A final remark

Our discussion shows why in quantum theory it makes sense to study representations not ofthe generic symmetry groups but of their corresponding universal covering group, which is (theunique) simply connected Lie group G

π−→ G covering G. Of course, every representation ρ

of the original Lie group G can be lifted to a representation of the universal covering, simplyby setting ρ := ρ π. The converse is not true in general, but every representation of theuniversal covering group defines a projective representation of the original symmetry group(cf. the example in 3.2.2.). For example, one might study the group SL(2,C), which is theuniversal covering group of the (proper orthochronous) Lorentz-group L+

↑ = SO(1, 3), or, moregenerally, the spin groups Spinn

π−→ SO(n) which can be realized as subgroups of the units ina Clifford algebra and lead to the so-called spin representations.

33

Chapter 4

The Central Extensions GLres(H) & Ures(H)

We construct a central extension GLres(H) of the restricted general linear group GLres(H)

by C× = C\0 which restricts to a central extension of Ures(H) by U(1). These groups actnaturally on the fermionic Fock space which will be constructed in the next chapter in the spiritof the Dirac sea theory. This action lifts the projective representation of Ures (respectively,GLres) on Gr(H), given by

Ures(H)×Gr(H)→ Gr(H),(U,W

)7−→ ρ(U)W = UW

(4.0.1)

to a proper representation on the Fock space. In this sense, the central extension can be under-stood as the “second quantization” of the groups GLres(H) and Ures(H). In more mathematicalterms, they represent an explicit construction of the “deprojectivization”, discussed in a moreabstract context in §3.3.1. (c.f. [Wurz01], Prop. II.27). One may think of them as “bigger”groups containing the information about the transformation itself, plus all possible choices forthe phase. A little more to the point, we can think of it in the following way: Ures(H) andGLres(H) act on Dirac seas only projectively. GLres(H) and Ures(H) contain the additional in-formation about how to rotate the basis of the Hilbert space appropriately, in order to turn thisinto a proper action on infinite-wedge-vectors (cf. the discussion of the left- and right-actionin Section 5.2). This provides a generalization of the product-wise lift

ϕ1 ∧ ϕ2 ∧ · · · ∧ ϕnU−→ Uϕ1 ∧ Uϕ2 ∧ · · · ∧ Uϕn

to states of infinitely many fermions.

4.1 The Central Extension of GLres

We start with the construction of the central extension of the identity component GL0res(H)

of GLres(H). The construction is based on the standard polarization H = H+ ⊕ H−, butapplication to a different (fixed) polarization is immediate. With respect to this splitting, wecan write every linear operator as

A =

(A++ A+−

A−+ A−−

)=

(a b

c d

).

35

4.1. THE CENTRAL EXTENSION OF GLRES

We defineE =

(A, q) ∈ GL0

res(H)×GL(H+) | a− q ∈ I1(H+).

Lemma 4.1.1 (Group structure of E).

E is a subgroup of GL0res(H)×GL(H+).

Proof. : Clearly, (IdH, IdH+) ∈ E . Furthermore:

i) If (A1, q1), (A2, q2) ∈ E with A1 =

(a1 b1

c1 d1

), A2 =

(a2 b2

c2 d2

), we have

(A1A2)++ = a1a2 + b1c2 and

(A1A2)++− q1q2 = a1a2+b1c2−q1q2 = (a1 − q1)︸︷︷︸∈I1(H+)

a2 + q1 (a2 − q2)︸︷︷︸∈I1(H+)

+ b1︸︷︷︸∈I2(H)

c2︸︷︷︸∈I2(H)

∈ I1(H+).

Thus: (A1A2, q1q2) ∈ E .

ii) Let (A, q) ∈ E .

A =

(a b

c d

)∈ GL0

res(H) ⇒ A−1 =

(α β

γ δ

)∈ GL0

res(H).

We deduce: aα+ bβ = IdH+⇒ aα ∈ Id+ I1(H+) and thus:

a (α− q−1) = aα︸︷︷︸∈Id+I1(H+)

− aq−1︸︷︷︸∈Id+I1(H+)

∈ I1(H+)

Since a is a Fredholm operator with index 0, it has a pseudoinverse. It follows that(α− q−1) ∈ I1(H+) and so (A−1, q−1) ∈ E .

Lemma 4.1.2 (Some properties of E).

i) a− q ∈ I1(H+) is equivalent to q−1a having a determinant.

ii) For all A ∈ GLres ∃ q ∈ GL(H+) such that (A, q) ∈ E.

iii) If aq−1 has a determinant then aq′−1 has a determinant if and only if q−1q′ does.

It follows that we can think of E as an extension (not a central one!) of GLres by GL1(H+):

1 −→ GL1(H+) −→ E pr2−−−→ GL0res(H) −→ 1

ii) implies that the projection pr2 is surjective onto GL0res(H) and iii) implies that two preim-

ages differ (multiplicatively) by an element in GL1(H+).

Proof. i) a− q ∈ I1(H+) ⇐⇒ q−1(a− q) = q−1a− Id ∈ I1(H+) ⇐⇒ q−1a ∈ Id+ I1(H+).ii) A ∈ GL0

res ⇒ U++|H+→H+=: a is a Fredholm operator of index 0. Thus, there exists a finite

rank operator t : H+ → H+ such that q := a− t is invertible and a− q = t ∈ I1(H+).iii) If q−1a and q′−1a both have determinants, so does (q−1a)(q′−1a)−1 = q−1q′.Conversely, if q−1a, q−1q′ have determinants, so does (q−1q′)−1(q−1a) = q′−1a.

36


Definition 4.1.3 (Central Extension of GLres).The group SL(H+) = q ∈ GL1(H+) | det(q) = 1 is embedded into E as

SL(H+) → E , q 7→ (1, q).

We defineGL

0

res(H) := E/ SL(H+)

and will show that this is a central extension of GL0res by C× := C \ 0.

More explicitely, GL0

res(H) is the quotient of E , modulo the equivalence relations

(A1, q1) ∼ (A2, q2) :⇐⇒ A1 = A2 and det(q−12 q1) = 1. (4.1.1)

Lemma 4.1.4 (Central Extension of GLres).The extension

1 −→ GL1(H+) −→ E pr2−−−→ GL0res(H) −→ 1

descends to a central extension

1 −→ C× ı−→ GL0

res(H)π−−→ GL0

res(H) −→ 1 (4.1.2)

with π : [(A, q)] 7→ A and ı : c 7→ [(Id, t)] for any t ∈ GL1(H+) with det(t) = c.

Proof. :

• Multiplication is well-defined on E/ ∼:If (A1, q1) ∼ (A1, r1) and (A2, q2) ∼ (A2, r2) i.e. det(q−1

1 r1) = 1 = det(q−12 r2) then

det(q−12 q−1

1 r1r2) = det(q−11 r1r2q

−12 ) = det(q−1

1 r1) det(r2q−12 ) = 1,

i.e. (A1A2, q1q2) ∼ (A1A2, r1r2).

• Now it’s easy to see that π and i are well-defined homomorphisms of groups and that πsurjective and i is injective.

• We show ker(π) = im(i): Note that ker(π) = (1, q) ⊂ E , with the q’s satisfying1++−q = 1H+

−q ∈ I1(H+) and thus q ∈ GL(H+)∩(Id+I1(H+)) = GL1(H+). Hence,ker(π) = im(i).

• The extension is central: For all (A, q) ∈ E and t ∈ GL1(H+), we find (A, qt) ∼ (A, tq),since det((qt)−1tq ) = det(t−1 q−1 t q) = det(t)−1 det(q−1 tq) = det(t)−1 det(t) = 1.Thus, [(A, q)] · [(1, t)] = [(1, t)] · [(A, q)] , ∀ [(A, q)] ∈ GL

0

res and [(1, t)] ∈ im(i).

37


4.1.1 The complete GLres

So far we have constructed the identity component GL0

res of the central extension, correspondingto transformations preserving the net-charge. Generalization to other connected componentsindexed by the relative charge q = charge(W,AW ) for A ∈ GLres,W ∈ Gr(H), is obtained byconcatenating charge-preserving transformations with a shift of a suitably chosen Hilbert-basis.

Let (ek)k∈Z be a basis of H such that (ek)k≥0 is a basis of H+ and (ek)k<0 a basis of H−. Weneed a σ ∈ GLres with ind(σ++) = ±1. Conveniently, we choose σ defined by ek 7−→ ek+1.This σ is unitary with ind(σ++) = −1. It acts on GL0

res by A 7→ σAσ−1. Now, we can definea semi-direct product on Z n GL0

res by

(n,A) ·n (m,A′) = (n+m,AσnA′σ−n). (4.1.3)

With this group-structure, the obvious map

Z n GL0res → GLres, (A,n) 7→ Aσn (4.1.4)

becomes an isomorphism of groups:

(n,A) · (m,A′) = (n+m,AσnA′σ−n) 7−→ AσnA′σ−nσn+m = AσnA′σm.

Thus, we can describe the restricted general group as a semi-direct product of the identitycomponent GL0

res with Z. The Z-component corresponds to (−1)× the index of the ++ com-ponent, i.e. to the relative charge “created” by the transformation.

The action of σ on GL0res (by conjugation) is covered by an endomorphism σ : E → E ,

σ((A, q)) = (σAσ−1, qσ), where

qσ =

σqσ−1 ; on σ(H+) = H≥1

Id ; on σ(H+)⊥ = span(e0)

This is not an automorphism on E , but it descends to an automorphism on GL0

res. With alittle abuse of notation, we write σ for this map as well.

Definition 4.1.5 (The complete GLres).We define

GLres(H) := Z n GL0

res(H)

with the action of Z on GL0

res(H) generated by σ. Then

1 −→ C× −→ GLres(H)π−−→ GLres(H) −→ 1

is a central extension of GLres by C× and restricts to (4.1.2) over the identity components ofthe Lie groups.

38


4.1.2 Lie Group- and Bundle- structure

We will show that GL0

res carries the structure of an infinite-dimensional Banach Lie group. Tothis end, we first introduce a Lie group structure on E by giving it not the subgroup topology,but the topology induced by the embedding

ι : E → Bε × I1(H+); (A, q) 7→ (A, a− q).

Lemma 4.1.6 (Lie Group structure of E).E with the topology defined by the embedding ι is a Banach Lie group.

Proof. Let A := Bε × I1(H+) be equipped with the product-norm

‖(A, t)‖A := ‖A‖ε + ‖t‖1.

We define a multiplication ? on A in such a way, as to make ι an algebra-homomorphism:

(A, t) ? (A′, t′) := (AA′, at′ + ta′ + bc′ − tt′).

It is easily checked that((A,+, ?), ‖·‖A

)is a Banach algebra with unit element (1, 0). We

claim that ι(ε) is precisely the group A× of unit elements in Bε× I1(H+) and therefore a linearBanach Lie group. Note that (A, t) ∈ A is invertible if and only if A ∈ GL1(H+)res(H) andthere exists t′ ∈ I1(H+) s.t.

(A, t) ? (A−1, t′) :=(1, at′ + ta′ + bc′ − tt′) = (1, 0)

for aa′ + bc′ = 1H+. And this is true, if and only if

(a− t)t′ + ta′ + bc′ = (a− t)t′ + ta′ + (1H+− aa′) = 0

⇐⇒ (t− a)t′ − (t− a)a′ = (t− a)(t′ − a′) = 1H+

⇐⇒ (A, t) = ι((A, t− a)

); with A ∈ GL1(H+)res(H), (t− a) ∈ GL1(H+).

This finishes the proof.

Now, we can understand GL0

res(H) as homogeneous space for the Lie subgroup SL(H+). Itfollows that there exist a unique Lie group structure on GL

0

res(H) = E/ SL(H+) such that thecanonical projection becomes a submersion ([Bou], §1.6, Prop. 11). We can conclude thatGL

0

res(H) is an infinite-dimensional Banach Lie group.

Bundle structure

In fact, π : GL0

res → GL0res is not only a central extension, it also carries the structure of a

principle fibre bundle. On principle-bundles, local trivializations are given by local sections. Aswe’re dealing we Lie groups, the bundle structure is already defined by a smooth section in aneighborhood of the identity (because any such section can be translated to arbitrary pointson the manifold by the multiplicative action of group on itself). For GL

0

res, there exists a verynatural local section about the identity in GL0

res(H) that will be of great importance for thefurther discussion.

39


ConsiderW := A ∈ GLres | a = A++ ∈ GL(H+) ⊂ GL0

res

Claim: W is a dense, open subset of GL0res(H).

Proof. Recall that the topology in GLres is given by the norm ‖·‖ε = ‖·‖∞ + ‖[ε, ·]‖2.Clearly, A 7→ a = P+AP+ is continuous w.r.to this norm, thereforeW is open in GLres becauseGL(H+) is open in the space of bounded operators on H+: For a ∈ GL(H+) and ‖k‖ small

enough, a− k is also invertible with (a− k)−1 =[ ∞∑ν=o

(a−1k)ν]a−1.

Furthermore, W is dense in GLres because for any Fredholm operator a with index 0 – such asa = A++ for A ∈ GLres – there exists a compact operator k of arbitrary small norm so thata+ k is invertible.

On W we can define a section

τ : W −→ GLres(H); A 7−→ [(A,A++)]. (4.1.5)

Obviously, τ is smooth because ι τ(A) = (A, 0) ∈ GLres × I1(H+). Also, τ(1) = 1. Thissection induces the local trivialization

φ : GLres(H) ⊇ W −→ GLres(H)× C×;

[(A, q)] 7−→(A ,det(a−1q)

) (4.1.6)

by φ−1(A, λ) := τ(A) · λ ∈ GLres(H).

Proposition 4.1.7 (Lie Group Cocycle).On W we compute the continuous 2-cocycle τ(A)τ(B) = χ(A,B) τ(AB) to be

χ(A,B) = det[A++B++(AB++)−1]. (4.1.7)

Proof. Let A,B ∈W s.t. AB ∈W . Then τ(AB) = [(AB, (AB) ++)]. And thus:

τ(A) · τ(B) =[(A,A++)] [(B,B++)] = [(AB,A++B++)]

=[(1, A++B++(AB++)−1)] τ(AB)

= det[A++B++(AB++)−1] · τ(AB)

From which we can immediately read off the cocycle.

The central extension of Lie groups

1 −→ C× ı−→ GL0

res(H)π−−→ GL0

res(H) −→ 1 (4.1.8)

induces a central extension0 −→ C ı−→ g1

π−−→ g1 −→ 0 (4.1.9)

of the Lie algebra. We compute the corresponding Lie algebra cocycle.

40


Proposition 4.1.8 (Lie Algebra Cocycle).The Lie algebra cocycle for to the central extension (4.1.9) is

c(X,Y ) = tr(X−+Y+− − Y−+X+−)

=1

4tr(ε [ε,X][ε, Y ])

(4.1.10)

This is known as the Schwinger cocycle or Schwinger term.

Proof. Using formula (3.5.5) we find:

c(X,Y ) =∂

∂t

∂

∂s

∣∣∣t=s=0

det[esX++ e

tY++((esXetY )++)−1

]− ∂

∂t

∂

∂s

∣∣∣t=s=0

det[etY++ e

sX++((etY esX)++)−1

]= tr

(X++Y++ − (XY )++

)− tr

(Y++X++ − (Y X)++

)= tr

(X++Y++ −X++Y++ −X+−Y−+

)− tr

(Y++X++ − Y++X++ − Y+−X+−

)= tr

(−X+−Y−+

)+ tr

(Y+−X−+

)The traces converge individually and put together we get

c(X,Y ) = tr(X−+Y+− − Y−+X+−

).

A straight forward computation shows that this can also be expressed as

c(X,Y ) =1

4tr(ε [ε,X][ε, Y ]).

4.1.3 The Central Extension of Ures and its Local Trivialization

Having defined the central extension GLres(H) of GLres(H), we set

Ures(H) :=

[(U, r)] ∈ GLres

∣∣(U, r) ∈ U(H)×U(H+)

(4.1.11)

This is a central extension of Ures(H) by U(1) and its complexification is GLres(H).

Ures(H) is a principle U(1)-bundle. But the section τ defined above does not restrict to asection in Ures(H) because for unitary U , U++ need not be unitary, even if it’s invertible. Wecan however use a polar decomposition to write

U++ = VU |U++| with VU ∈ U(H+) unitary, |U++| =√U∗++U++. (4.1.12)

Lemma 4.1.9 (Local Section of Ures).On W ∩Ures(H), the map

σ : U 7−→ [(U, VU )] (4.1.13)

defines a local section in Ures(H). The local trivialization φUres induced by this section equalsφ up to normalization.

41


Proof. For U ∈ Ures, unitarity implies U∗++U++ + U∗+−U+− = 1H+. As the odd-parts are

Hilbert-Schmidt operators, U∗+−U+− is trace-class. Therefore, U∗++U++ ∈ IdH+ + I1(H+) hasa determinant. It follows that

√(U∗++U++) = |U++| is also in IdH+

+I1(H+) and we conclude

U++ − VU = VU(|U++| − 1H+

)∈ I1(H+).

Hence, [(U, VU )] ∈ Ures(H) for all U ∈ Ures. Furthermore, we find φ([(U, r)]

)= (U, λ) with

λ = det(U−1++ r) = det(|U++|−1) det(V ∗U r) = det

(√(U−1

++ r)(U−1++ r)

∗)

det(V ∗U r)

=

√det(U−1

++ r) det(U−1++ r) det(V ∗U r) = |λ|det(V ∗U r).

But det(V ∗U r) is just the U(1)-component w.r.to the local trivialization φUresdefined by σ.

Hence we read off:

U ∈ Ures(H)

φ

vv

φUres

((Ures × C× 3 (U, λ)

(U, λ|λ|

)∈ Ures ×U(1)

(4.1.14)

In particular, for r = VU we find φσ(U) =(U, 1|det(U++)|

)∈ Ures×C×, showing that the local

section σ is smooth in GLres(H) (see Prop. 5.3.6 for smoothness of the determinant). Thisfinishes the proof.

In the local trivialization of Ures(H), the U(1)-component corresponds to the information aboutthe phase of the lift of the unitary transformation. In this sense, we can think of the sectionsas “gauging” the phases of the lifts by picking out a reference lift.

It can be readily computed that the cocycle for the section σ is, as one would expect, just thenormalized version of the GLres-cocycle χ in (4.1.7), i.e.

χU(U, V ) := det[U++V++(UV++)−1]/ |det[U++V++(UV++)−1] |. (4.1.15)

For the corresponding central extension of Lie algebras, 0 −→ R −→ ures −→ ures −→ 0, wecompute the cocycle

∂

∂t

∂

∂s

∣∣∣t=s=0

χ(esX , etY ) |χ(esX , etY )|−1 − ∂

∂t

∂

∂s

∣∣∣t=s=0

χ(etY , esX) |χ(etY , esX)|−1

=∂

∂t

∂

∂s

∣∣∣t=s=0

[χ(esX , etY )

(χ(esX , etY )χ(esX , etY )

)− 12 − χ(etY , esX)

(χ(etY , esX)χ(etY , esX)

)− 12

]= c(X,Y )− 1

2

(c(X,Y ) + c(X,Y )

)= i Im

(c(X,Y )

), for X,Y ∈ ures.

But as all X,Y ∈ ures are (anti-)Hermitian, we find

c(X,Y ) = c(Y ∗, X∗) = c(Y,X) = −c(X,Y ),

which means i Im(c(X,Y )

)= c(X,Y ) on ures. Thus, the Lie algebra cocycle on ures defined

by σ is just the Schwinger cocycle restricted to the subalgebra ures ⊂ g1.

42

4.2. NON-TRIVIALITY OF THE CENTRAL EXTENSIONS

4.2 Non-Triviality of the Central Extensions

Theorem 4.2.1 (Non-triviality of the Central Extension).The following equivalent statements hold true:

i) The central extension GLres(H)π−−→ GLres(H) and Ures(H)

π−−→ Ures(H) are not trivial.

ii) The Lie algebra extensions g1 → g1 and ures → ures are not trivial.

iii) The Schwinger cocycle (4.1.10) is not trivial in H2(g1,C), respectively in H2(ures, iR).

iv) There exists no continuous (local) section Γ : Ures(H) → Ures(H) which is also a homo-morphism of groups. The same is true for GLres(H).

v) There exist no unitary representation of Ures(H) on the fermionic Fock space, lifting theprojective representation (4.0.1). The analogous statement is true for GLres(H).

Proof of the Theorem. 1 For the equivalence of the statements, we have to recall the discussionin Chapter 3. Note that any continuous Lie group homomorphism ϕ between linear Banach Liegroups induces a unique continuous homomorphism Lie(ϕ) = ϕ between their Lie algebras withexp Lie(ϕ) = ϕ exp (see Appendix A.2). Thus, a continuous splitting map for the centralextension of Lie groups would induce a splitting map for the central extension of Lie algebras,showing ii) ⇒ i) ⇐⇒ iv). i) ⇒ ii) is Prop. 3.5.10 [Triviality of extensions]. ii) ⇐⇒ iii)

is Thm. 3.5.6 [Triviality of Lie algebra extensions]. Finally, i) ⇐⇒ v) follows analogouslyto Prop. 3.3.6 [Lifting projective representations] from the representation of GLres(H) onthe fermionic Fock space that will be constructed in the next chapter. Def. 5.3.14 will givean embedding of Gr(H) into the projective Fock space P(F), known as the Pflücker embedding.

Now, we are actually going to prove iii), i.e. show that the Schwinger cocycle c is not trivial.This means that there is no linear map µ : ures → R with µ([X,Y ]) = c(X,Y ), ∀X,Y ∈ ures.This is proven, for example, if we can find X,Y ∈ ures with [X,Y ] = 0 but c(X,Y ) 6= 0.

For completeness, we recall the argument why this proves that there exists no continuous sectionin Ures(H) which is also a homomorphism. In Lemma 4.1.9, we have constructed a smoothsection σ : Ures → Ures. However, this section fails to be a homomorphism of groups, i.e. weget σ(U)σ(V ) = κ(U, V )σ(UV ) with a Lie group cocycle κ : Ures×Ures → U(1). Consequently,its derivative (at the identity) σ : ures → ures fails to be a Lie algebra homomorphism. This isexpressed by the Schwinger cocycle

c(X,Y ) := [σ(X), σ(Y )]− σ([X,Y ]) for X,Y ∈ ures.

Now suppose the section σ is just a bad choice and there was in fact a different section Γ :

Ures → Ures which is also a Lie group homomorphism. Since two elements in the same fibre inUres differ only by a complex phase, the “good” section differs (multiplicatively) from σ by amap λ : Ures → U(1). As Γ is a Lie group homomorphism, the corresponding Lie algebra mapΓ = λ+ σ is a continuous Lie algebra homomorphism.

1The proof follows [Wurz01], where it is presented in a very nice and complete way. However,we believe that the abstract mathematical language might conceal the ultimately simple nature ofthe proof, at least from a physicist’s point of view. We have therefore tried to rephrase it in moreelementary terms, without any reference to cohomology or the like.

43


This means:

0 = [λ(X) + σ(X), λ(Y ) + σ(Y )] = [σ(X), σ(Y )] = λ([X,Y ]) + σ([X,Y ])

⇒ λ([X,Y ]) = c(X,Y ), ∀X,Y ∈ ures.

For the first equality we have used that λ : ures → Lie(U(1)) = R maps into the centerof ures and so all the commutators with λ vanish. We observe: if a lift Γ : Ures → Ures

preserving the group structure exists, then there exists a linear map µ (= λ) : ures → R withµ([X,Y ]) = c(X,Y ), ∀X,Y ∈ ures. Therefore, to prove that such a lift does not exist, itsuffices to find X,Y ∈ ures with

[X,Y ] = 0 but c(X,Y ) 6= 0.

This would prove the statement for both the unitary and the general linear case.

The Hilbert space H = H+ ⊕ H− with the polarization given by the sign of the free DiracHamiltonian is somewhat difficult to handle. Fortunately, by unitary equivalence we can justas well consider any other polarized (separable, infinite-dimensional, complex) Hilbert space.For our purpose it is nice to work with the Hilbert space K := L2(S1,C) with the polarizationgiven by separation into Fourier components with positive and negative frequencies. Thenatural Hilbert-basis on K is the Fourier-Basis (ek)k∈Z where ek(t) := ei2πkt ∈ L2(S1,C).

Writing K 3 f =∑k∈Z

fkek we have:

P+f :=∑k≥0

fkek , P−f :=∑k<0

fkek (4.2.1)

On K we consider the class of operators given by multiplication with smooth functions. Forg ∈ C∞(S1,C) and f ∈ L2(S1,C) we write Mg(f) = g · f .In Fourier space, multiplication corresponds to convolution. So, if g =

∑k∈Z

gk ek then

Mg(el) =∑k∈Z

gk−l ek (4.2.2)

Be careful not to make the easy mistake to confuse the Fourier components of g with those ofthe multiplication operator Mg : K → K.For g, h ∈ C∞(S1,C) we compute:

tr(Mh−+Mg+−) =∑l<0

〈el,Mh

∑k≥0

gk−lek〉 =∑l<0

〈el,∑k≥0

∑m<0

hm−k gk−l em〉

=∑l<0

∑k≥0

hl−k gk−l = −∑l<0

l hl g−l. (4.2.3)

Similarly:tr(Mg−+Mh+−) =

∑l>0

l hl g−l. (4.2.4)

44


In particular, we conclude that

‖[ε,Mg]‖22 =4 · tr(

(Mg+−)(Mg+−)∗ + (Mg−+)(Mg−+)∗)

= 4 ·(∑l≥0

l|gl|2 +∑l<0

(−l)|gl|2)

= 4 ·(∑l∈Z|l||gl|2

)<∞ for smooth g.

So all operators of this type have off-diagonal components in the Hilbert-Schmidt class, meaningthat indeed

Mg

∣∣ |g(t)|2 ≡ 1⊂ Ures(K) ;

Mg

∣∣ g(t) ∈ R⊂ ures(K).

From (4.2.3) and (4.2.4) we also read off that the Schwinger cocycle is well-defined on operatorsof this type with

c(Mh,Mg) = tr(Mh−+Mg+− −Mg−+Mh+−) = −∑l∈Z

lhlg−l

=1

2πi

1∫0

h(t)g(t) dt (4.2.5)

The rest is easy. Just take any two functions g, h ∈ C∞(S1,R) with1∫0

h(t)g(t) dt 6= 0.

For instance, consider g(t) = cos(2πt) and h(t) = sin(2πt). Then we got

Mg,Mh ∈ ures(h) with [Mh,Mg] = 0 but

c(Mh,Mg) =1

2πi

1∫0

h(t)g(t) dt = (−1)

1∫0

sin2(t) dt 6= 0

This shows that the Schwinger cocycle c is non-trivial and completes our proof.

In fact, we have proven a much stronger result that doesn’t require continuity at all.

Theorem 4.2.2 (Non-Triviality, algebraic version).There exist no local homomorphism τ : Ures(H)→ Ures(H) with π τ = Id.Consequently, there exists no homomorphism ρ : Ures(H) → U(F) that lifts the projectiverepresentation (4.0.1), not even locally. The same is true for GLres(H).

Proof. Again, we present the proof for the unitary case. The general linear case follows anal-ogously. Recall that for the central extension

1 −→ U(1)ı−→ U0

res(H)π−−→ U0

res(H) −→ 1

we have the 2-cocycle (4.1.15)

χU(U, V ) := det[U++V++(UV++)−1]/|det[U++V++(UV++)−1]|

45


coming from the smooth local section (4.1.13). Suppose there was an open neighborhood O ofthe identity in Ures(H) and a homomorphism τ : O → Ures(H) with π τ = Id. Then, χ wastrivial as a (local) algebraic group 2-cocycle, i.e. τ would give rise to a map λ : O → U(1) with

λ(UV ) = χ(U, V )λ(U)λ(V ), ∀U, V ∈ O

(see the discussion at the beginning of §. 3.4). Now, we can proof that such a map cannotexist anywhere near 1 ∈ Ures(H), by showing that every open neighborhood contains elementsU, V with UV = V U , but χ(U, V ) 6= χ(V,U).

But in fact, this was already shown in the proof of the previous theorem, where we foundX,Y ∈ ures = Lie(Ures(H)) with [X,Y ] = 0 and c(X,Y ) 6= 0, where c : ures → iR wasthe Schwinger term, i.e. the Lie algebra cocycle corresponding to χ. Because given suchX,Y ∈ ures, [X,Y ] = 0 implies [exp(sX), exp(tY )] = 0 in Ures(H) for all s, t ∈ R sufficientlysmall. And using the identity (3.5.5) i.e.

c(X,Y ) =∂

∂t

∂

∂s

∣∣∣t=s=0

χU(esX , etY ) − ∂

∂t

∂

∂s

∣∣∣t=s=0

χU(etY , esX)

we see that ∀ε > 0 ∃s, t ∈ (0, ε) : χ(esX , etY ) 6= χ(etY , esX), since the contrary would implyc(X,Y ) = 0. Note that we use continuity or differentiability only of the section σ, which allowsus to apply (3.5.5). The final result, however, is purely algebraical.

Remark 4.2.3. (Embedding of Loop Groups)In the related mathematical literature, the multiplication operators and the cocycle (4.2.5)considered in the proof of the previous theorem arise in the abstract context of embeddings ofloop groups into GLres(H). If K is a d-dimensional (compact) Lie group, then C∞(S1,K) is aninfinite-dimensional Lie group, called a loop group and usually denoted by LK orMap(S1,K).Any representation ρ : K → GL(Cd) then provides an action of the loop group on the Hilbertspace K = L2(S1,C) via

C∞(S1,K)× L2(S1,C) = LK ×K → K

(ϕ, f) 7−→ ρ(ϕ(t)) · f(t)

Just as we did above (for d=1), this provides an embedding of the loop group LK intoGLres(K) ∼= GLres(H). The central extension GLres of GLres induces a central of LK andthe corresponding Lie algebra cocycle takes a form analogous to (4.2.5). Multiplication in Cthen is just replaced by matrix multiplication and taking the trace in Mat(d × d,C). In ourproof we have explicitly avoided any reference to loop groups or cohomology theory as usu-ally found in the mathematical literature, since we feel that they unnecessarily obscure theotherwise simple nature of the proof.

46

Chapter 5

Three Routes to the Fock space

5.1 CAR Algebras and Representations

In this section we carry out the quantization of the Dirac field in the spirit of the “electron-positron picture”. We assume that the reader is somewhat familiar with creation/annihilationoperators and the standard construction of the Fock space and review them just briefly. Therather hands-on construction is followed by a brief discussion of abstract C∗ - and CAR alge-bras. Fock spaces will then arise as representation spaces of irreducible representations of theCAR-algebra. This is quite an abstract mathematical machinery, but I think it’s rewarding fordifferent reasons:

1. It is probably the most common and most developed mathematical description of Fockspaces and “second quantization”.

2. It constitutes, at least rudimentary, a rigorous mathematical formalization of what physi-cists are usually trying to say.

3. It provides a language in which the mathematical problems can be formulated in a veryprecise way and reveals an abstract perspective that can be fruitful from time to time.

What do we mean by a “fruitful perspective”? For instance, as a physicist with some train-ing in (non-relativistic) quantum theory one is often used to think of “the” Fock space as afundamental object (“the space of all physical states”). This makes it hard to grasp some ofthe issues we’re facing in relativistic quantum field theory, for example the fact that the timeevolution is “leaving” the Fock space (where else would it go?). Thus it can be helpful to thinkof the CAR-algebra as the more fundamental object and of different Fock spaces correspondingto different representations. The danger, however, is that this easily becomes a vain exercisein abstract mathematics, detached from the physical problems.

5.1.1 The Field Operator

On the one-particle Hilbert-space H = H+ ⊕ H− we have a charge conjugation operator Cmapping negative energy solutions of the free Dirac equation to positive energy solutions withopposite charge. The charge-conjugation is anti-unitary, in particular anti-linear. The exactform of the operator depends on the representation of the Dirac algebra. In the so-called“Majorana representation”, charge conjugation just corresponds to complex conjugation. InAppendix A.2 we give an intuitive, yet general derivation of the charge-conjugation operator.

47

5.1. CAR ALGEBRAS AND REPRESENTATIONS

Definition 5.1.1 (Fermionic Fock space).Let F+ := H+ and F− := CH−. We define the fermionic Fock space as 1

F :=

∞⊕n,m=0

F (n,m); F (n,m) :=∧nF+ ⊗

∧mF− (5.1.1)

We can split the Fock space into the different charge sectors

F :=

∞⊕c=0

F (c); F (c) :=⊕

n−m=c

F (n,m). (5.1.2)

The stateΩ := 1⊗ 1 ∈ F (0,0) = C⊗ C

is called the vacuum state.

Recall the definitions of the creation and annihilation operators.On the “particle-sector”

∧F+:

a(f) : F (n+1,m) −→ F (n,m),

a(f)f0 ∧ · · · ∧ fn :=

n∑k=0

(−1)k〈f, fk〉f0 ∧ · · · ∧ fk ∧ · · · ∧ fn (5.1.3)

a∗(f) : F (n−1,m) −→ F (n,m),

a∗(f)f1 ∧ · · · ∧ fn−1 = f ∧ f1 ∧ · · · ∧ fn−1 (5.1.4)

On the “anti-particle sector”∧F−:

b(g) : F (n,m+1) −→ F (n,m),

b(g) Cg0 ∧ · · · ∧ Cgn := (−1)nn∑k=0

(−1)k〈Cg, Cgk〉Cg0 ∧ · · · ∧ Cgk ∧ · · · ∧ Cgn (5.1.5)

b∗(g) : F (n,m−1) −→ F (n,m),

b∗(g) Cg1 ∧ · · · ∧ Cgn−1 = Cg ∧ Cg1 ∧ · · · ∧ Cgn−1 (5.1.6)

The reader is probably familiar with the fact that a and a∗, as well as b and b∗, are formaladjoints of each other and satisfy the canonical anti-commutation relations

a(f1), a∗(f2) = a(f1)a∗(f2) + a∗(f2)a(f1) = 〈f1, f2〉H · 1, ∀f1, f2 ∈ H+

b(g1), b∗(g2) = 〈Cg1, Cg2〉H · 1 = 〈g1, g2〉 · 1 = 〈g2, g1〉H · 1, ∀g1, g2 ∈ H−(5.1.7)

and all other possible combination anti-commute.

If (fj)j∈N and (gk)k∈N are ONB’s of H+ and H− respectively, the elements of the form

a∗(fj1)a∗(fj2) . . . a∗(fjn)b∗(gk1)b∗(gk2

) . . . b∗(gkm)Ω ∈ F (n,m) ⊂ F (5.1.8)

for j1 < · · · < jn; k1 < · · · < km and n,m = 0, 1, 2, 3, . . . form an ONB of the Fock space F .

1By the direct sums we implicitly understand the completion w.r.to the induced scalar product.

48


So, the creation operators acting on the vacuum generate the dense subspace

D :=finite linear combinations of vectors of the form (5.1.8)

which is the convenient domain for the second quantization of bounded operators.

Definition 5.1.2 (Field Operator).

For any f ∈ H we define the field operator Ψ(f) on F by

Ψ(f) := a(P+f) + b∗(P−f)

Ψ∗(f) = a∗(P+f) + b(P−f)(5.1.9)

We can view the field operator as an anti-linear map Ψ : H −→ B(F).

It satisfies the canonical anti-commutation relations (CAR)

Ψ(f),Ψ∗(g)

= 〈f, g〉 · 1

Ψ(f),Ψ(g)

=

Ψ(f)∗,Ψ∗(g)

= 0(5.1.10)

Second Quantization of Unitary Operators

Let U : H → H be a unitary transformation on the one-particle Hilbert space. We want tolift it to a unitary transformation Γ(U) on the Fock space F . In non-relativistic quantummechanics, we would lift an operator U to the n-particle anti-symmetric Hilbert-space

∧nH

“product-wise”, i.e. toU ⊗ U ⊗ · · · ⊗ U. (5.1.11)

But the naive generalization

f1 ∧ · · · ∧ fn ⊗ Cg1 ∧ · · · ∧ Cgm 7−→ Uf1 ∧ · · · ∧ Ufn ⊗ CUg1 ∧ · · · ∧ CUgm

make sense only if U preserves the splitting H = H+ ⊕ H− i.e. only if U+− = U−+ = 0. Ingeneral this is not the case and U will mix positive and negative energy states. Physically, thisleads to the phenomenon we call pair creation. Mathematically, this leads to trouble.

While it is difficult to say how the unitary transformation is supposed to act on the Fock space,there is a very natural action on the field operator Ψ, given by

Ψ 7−→ βu(Ψ) := Ψ U,

i.e. βu(Ψ)(f) =Ψ(Uf) = a(P+Uf) + b∗(P−Uf), ∀f ∈ H.(5.1.12)

βU is called a Bogoljubov transformation. It is easy to see that Ψ := βU (Ψ) is still an antilinearmap H → B(F) satisfying the CAR (5.1.10). This means that Ψ is also a field operator,inducing another representation of the CAR-algebra with the new annihilation operators definedas

c(f) := Ψ(f), for f ∈ H+

d(f) := Ψ∗(g), for g ∈ H−.(5.1.13)

49


Definition 5.1.3 (Implementability of Unitary Transformations).The unitary transformation U ∈ U(H) is implementable on the Fock space F if there exists aunitary map Γ(U) : F → F with

Γ(U)Ψ(f)Γ(U)∗ = βU (Ψ)(f) = Ψ(Uf), ∀f ∈ H. (5.1.14)

If U is implementable, the implementation is unique up to a phase.

That the implementation can be determined only up to a phase is obvious, because if Γ(U) isan implementation of U ∈ U(H), so is eiϕ Γ(U) for any eiϕ ∈ U(1).Supposed that Γ(U) is an implementation of U , we note that by (5.1.14) it acts on a basisvector of the form

a∗(f1)a∗(f2) . . . a∗(fn)b∗(g1)b∗(g2) . . . b∗(gm) Ω

= Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(g1)Ψ(g2) . . .Ψ(gm) Ω

in the following way

Γ(U)(

Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(g1) . . .Ψ(gm) Ω)

= Γ(U)Ψ∗(f1)Γ(U)∗Γ(U)Ψ∗(f2)Γ(U)∗ . . .Γ(U)Ψ(gm−1)Γ(U)∗Γ(U)Ψ(gm)Γ(U)∗Γ(U) Ω

= Ψ∗(Uf1)Ψ∗(Uf2) . . .Ψ∗(Ufn)Ψ(Ug1) . . .Ψ(Ugm)[Γ(U) Ω

]= c∗(f1)c∗(f2) . . . c∗(fn)d∗(g1)d∗(g2) . . . d∗(gm) Ω

with Ω = Γ(U)Ω. This simple consideration yields an important result:

Proposition 5.1.4 (New Vacuum).A unitary transformation U ∈ U(H) is implementable on F , if and only if there exists a vacuumfor the new annihilation operators (5.1.13) defined by βU (Ψ) in the same Fock space. That is,if there exists a normalized state Ω ∈ F with

c(f)Ω = βU (Ψ)(f) Ω = 0, ∀f ∈ H+ (5.1.15)

d(g)Ω = βU (Ψ)∗(g) Ω = 0, ∀g ∈ H−. (5.1.16)

Proof. If Γ(U) is an implementer of U set Ω = Γ(U)Ω. Then,

βU (Ψ)(f)Ω = Γ(U)Ψ(f)Γ(U)∗Γ(U)Ω = Γ(U)Ψ(f)Ω = 0, ∀f ∈ H+ (5.1.17)

βU (Ψ)∗(g)Ω = Γ(U)Ψ(g)Γ(U)∗Γ(U)Ω = Γ(U)Ψ(g)Ω = 0, ∀g ∈ H− (5.1.18)

hence Ω is a vacuum for βU (Ψ). Conversely, if there exists a vacuum Ω for βU (Ψ), we candefine a unitary transformation on F by

a∗(f1)a∗(f2) . . . a∗(fn)b∗(g1)b∗(g2) . . . b∗(gm) Ω

7−→ eiφc∗(f1)c∗(f2) . . . c∗(fn)d∗(g1)d∗(g2) . . . d∗(gm) Ω

with an arbitrary phase eiφ and the computation above shows that this is an implementationof U on F .

50


In the simplest case, when U leaves the positive and negative energy subspace invariant, wenote that c(f) = a(Uf), ∀f ∈ H+ and d(g) = b(Ug), ∀g ∈ H−, so that Ω = eiϕΩ. Of coursein this case, the phase is conveniently chosen to be 1. Consequently, the implementation isnothing else than

f1 ∧ · · · ∧ fn ⊗ Cg1 ∧ · · · ∧ Cgm = a∗(f1) . . . a∗(fn)b∗(g1) . . . b∗(gm) Ω

Γ(U)7−→ c∗(f1) . . . c∗(fn)d∗(g1) . . . d∗(gm) Ω = Uf1 ∧ · · · ∧ Ufn ⊗ CUg1 ∧ · · · ∧ CUgm.(5.1.19)

We see that the Bogoljubov transformation of the field operator and its implementation (if itexists) are indeed the natural generalization of the product-wise lift of U to the Fock-space.Almost sneakily, we have established a duality in our thinking about the implementation prob-lem that is quite exciting. Either we can formulate the problem as one about lifting a unitarytransformation U on H to the Fock space F , as was our initial motivation. Or we can ask theequivalent question, whether the two representations of the CAR-algebra induced by Ψ andβU (Ψ), respectively, are equivalent. So far, we have but hinted at this fact. A more detaileddiscussion will follow in the next section.

We can finally state our first version of the Shale-Stinespring theorem.

Theorem 5.1.5 (Shale-Stinespring, explicit version). 2

A unitary operator U ∈ U(H) is unitarily implementable on F if and only if the operators U+−

and U−+ are Hilbert-Schmidt. In this case, an implementation Γ(U) acts on the vacuum as

Γ(U) Ω := NeiφL∏l=1

a∗(fl)

M∏m=1

b∗(gm) eAa∗b∗ Ω (5.1.20)

where

• eφ ∈ U(1) is an arbitrary phase

• N =√

det(1− U+−U∗−+) =√

det(1− U−+U∗+−) is a normalization constant3

• A := (U+−)(U−−)−1 is Hilbert-Schmidt

• f1, . . . , fL is an ONB of kerU∗++ and g1, . . . , gM an ONB of kerU∗−−.

Recall that U++ and U−− are Fredholm-operators, therefore kerU∗++ and kerU∗−− are in-deed finite-dimensional subspaces. Also, (U−−)−1 is well-defined and bounded on imU−− =

(kerU∗−−)⊥ and we extend it to 0 on kerU∗−−. In this way, the operator A := (U+−)(U−−)−1

is well-defined.

In the explicit form of the transformed vacuum our result (2.2.8) about the net-charge "created"by U becomes manifest. From (5.1.20) we can directly read of that Γ(U)Ω contains M =

dim ker(U∗−−) particles of negative charge and L = dim ker(U∗++) particles of positive charge.

2see [Tha], Thms. 10.6 & 10.73Note that, in contrast to [Tha], we use the notation U∗

−+ = (U∗)−+ and not (U−+)∗ = (U∗)+−

51


Hence the net charge is

L−M = dim ker(U∗++)− dim ker(U∗−−)

= dim ker(U−−)− dim ker(U∗−−)

= ind(U−−) = − ind(U++).4

(5.1.21)

For the last equality we need a Lemma, which is proven in Appendix A.3.

The different sign in comparison to (2.2.8) comes from the unfortunate fact that the roles ofH+ and H− are interchanged in the two chapters.

Second Quantization of Hermitian Operators

Now we’re looking at the second quantization of an arbitrary bounded operator A : H → H,which is usually denoted by dΓ(A). Self-adjoint operators, as generators of the unitary group,are of particular interest. To

∧nH, we would lift an operator A additively as:

A⊗ 1⊗ · · · ⊗ 1 + 1⊗A⊗ 1 . . .1 + · · ·+ 1⊗ · · · ⊗ 1⊗A (5.1.22)

But again, the immediate generalization to F =∧H+⊗

∧C(H−) makes sense only if A respects

the polarization. So again, we should look at the field operator to formulate a generalizationof the additive lifting prescription. We will demand the following conditions for the “secondquantization”:

[dΓ(A),Ψ∗(f)] = Ψ∗(Af)

[dΓ(A),Ψ(f)] = −Ψ(A∗f)(5.1.23)

The minus-sign in the second line might look ad-hoc, but its relevance will become apparentsoon. Note that if dΓ(A) satisfies (5.1.23) so does dΓ(A) + c1F for every constant c ∈ C.Therefore, the implementation of a Hermitian operator, if it exists, is well defined only up toa constant. An operator dΓ(A) satisfying (5.1.23) then acts on a basis state (5.1.8) as

dΓ(U)(

Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(g1) . . .Ψ(gm)Ω)

=

n∑j=1

Ψ∗(f1) . . .Ψ∗(fj−1)Ψ∗(Afj)Ψ∗(fj+1) . . .Ψ∗(fn) Ψ(g1) . . .Ψ(gm) Ω

−m∑k=1

Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn) Ψ(g1) . . .Ψ(gk−1)Ψ(A∗gk)Ψ(gk+1) . . .Ψ(gm) Ω

+ Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn) Ψ(g1)Ψ(g2) . . .Ψ(gm)(dΓ(A)Ω

)We see that dΓ(A) is well-defined on the dense domain D if and only if the last summanddΓ(A)Ω is well-defined. In this case, dΓ(A) can be regarded as a generalization of the additive-lift (5.1.22). But how do we find such an operator dΓ(A) on F , satisfying 5.1.23?

Let (ek)k∈Z be a basis of H, s.t. (ek)k≤0 is a basis of H− and (ek)k>0 a basis of H+.A first educated guess for the second quantization of a bounded operator A would be

52


dΓ′(A) := AΨ∗Ψ, where AΨ∗Ψ is short-hand notation for5∑i,j∈Z〈ei, Aej〉Ψ∗(ej)Ψ(ei).

This expression is easily shown to be independent of the choice of basis. We can decompose itinto creation and annihilation operators and write AΨ∗Ψ = Aa∗a+Aa∗b∗ +Aba+Abb∗. Thevacuum expectation value of this operator can then be computed as

〈Ω,dΓ′(A)Ω〉 = 〈Ω, AΨ∗ΨΩ〉 = 〈Ω, Abb∗Ω〉

=∑i,j<0

〈ei, Aej〉〈Ω, b(ei)b∗(ej)Ω〉

=∑i,j<0

〈ei, Aej〉 δij = tr(A−−).

So unless A−− is trace-class, not even the vacuum state is in the domain of AΨ∗Ψ and thus,A more promising attempt is therefore

dΓ(A) = : AΨ∗Ψ : = Aa∗a+Aa∗b∗ +Aba−Ab∗b (5.1.24)

which also satisfies (5.1.23). Physicists call this normal ordering and denote it by two colonsembracing expressions like AΨ∗Ψ. With this prescription, : AΨ∗Ψ : Ω = Aa∗b∗Ω and therefore

〈Ω,dΓ(A) Ω〉 = 〈Ω, : AΨ∗Ψ : Ω〉 = 0. (5.1.25)

This implies immediately: AΨ∗Ψ : = AΨ∗Ψ− tr(A−−) · 1 (5.1.26)

whenever |tr(A−−)| <∞, so that in this case, AΨ∗Ψ and : AΨ∗Ψ : are both second quantiza-tions of A differing by a complex constant (or a real constant, if A is self-adjoint).

Assuming that A+− : H− → H+ is compact, there exists an ONB (uj)j of H+ and (vj)j of

H− and singular values λj ≥ 0 such that A+− =∞∑j=0

λj〈vj , ·〉uj .6

Hence, Aa∗b∗ =∑j

λja∗(uj)b

∗(vj) and we compute

‖Aa∗b∗Ω‖2 =∑k,j

λkλj〈Ω, b(vk)a(uk)a∗(uj)b∗(vj)Ω〉

=∑k,j

λkλj〈Ω, b(vk)(a(uk), a∗(uj) − a∗(uj)a(uk)

)b∗(vj)Ω〉

=∑k,j

λkλj

(δjk〈Ω,

(b(vk), b∗(vj) − b∗(vj)b(vk)

)Ω〉 − 〈Ω, b(vk)a∗(uj)b

∗(vj)a(uk)Ω〉)

=∑k,j

λkλjδjk =∑k

λ2k = ‖A+−‖22

This shows that : AΨ∗Ψ : is well-defined on the dense domain D if and only if A+− is ofHilbert-Schmidt type. For a Hermitian operator, the same must be true for the adjoint so werequire A−+ to be Hilbert-Schmidt as well.

5Consult e.g. [Tha] §10 for more details.6λ2

j are the eigenvalues of the positive operator (A+−)∗(A+−).

53


How does this relate to the discussion in the previous section? The relationship between uni-tary and self-adjont operators is claryfied by Stone’s Theorem, which says that every stronglycontinuous, one-parameter group of unitary operators is generated by a unique self-adjointoperator. We are in particular interested in the free time evolution Ut = e−itD0 .Let Ψt := βUt(Ψ) be the time-evolving field operator. Then, for any fixed f in the domain ofD0, Ψt(f) is even norm-differentiable in t because

limh→0

∥∥ 1

h[Ψt+h(f)−Ψt(f)]−Ψt(−iD0 f)

∥∥ = limh→0

∥∥Ψ(eiD0t[

1

h(eiD0h − 1) + iD0]f

)∥∥= limh→0

∥∥[ 1

h(eiD0h − 1) + iD0

]f∥∥ = 0

and thusd

dtΨt(f) = Ψt(−iD0 f) = iΨt(D0 f). (5.1.27)

The same is true for Ψ∗t , but note that Ψ∗t is now linear and not anti-linear in f.On a basis state of the form (5.1.8) we therefore compute

id

dt

∣∣∣t=0

[Γ(Ut)

(Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(g1) . . .Ψ(gm) Ω

)]= i

d

dt

∣∣∣t=0

[Ψ∗t (f1)Ψ∗t (f2) . . .Ψ∗t (fn)Ψt(g1) . . .Ψt(gm) Ω

]= Ψ∗(D0f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(g1) . . .Ψ(gm)Ω + Ψ∗(f1)Ψ∗(D0f2) . . .Ψ∗(fn)Ψ(g1) . . .Ψ(gm)Ω + · · ·

+ Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(D0fn)Ψ(g1) . . .Ψ(gm)Ω−Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(D0g1) . . .Ψ(gm)Ω− · · ·

− Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(g1) . . .Ψ(D0gm)Ω

= dΓ(D0)(

Ψ∗(f1)Ψ∗(f2) . . .Ψ∗(fn)Ψ(g1) . . .Ψ(gm) Ω)

This means:id

dtΓ(e−iD0t) = dΓ(D0) = : D0Ψ∗Ψ : (5.1.28)

on the dense domain where both sides are well-defined. The operator dΓ(D0) is called thesecond quantization of the free Dirac Hamiltonian.

If (fk)k∈N is a basis of H+ and (gk)k∈N a basis of H− then

dΓ(D0) = : D0Ψ∗Ψ :

=

∞∑i=1

∞∑j=1

(〈fi, D0fj〉a∗(fi)a(fj)− 〈gj , D0gi〉b∗(gi)b(gj)

) (5.1.29)

which is obviously semi-bounded from below because −D0 is positive definite on H−.Note that the anti-linearity of the charge-conjugation and thus of the Fock-space sectors

∧F−

makes all the difference. It is also the origin of the minus-sign in (5.1.23). Similarly, the globalgauge transformations e−it1 are generated by the charge-operator

Q := dΓ(1) =

∞∑i=1

∞∑j=1

(a∗(fi)a(fj)− b∗(gi)b(gj)

)(5.1.30)

which is just the difference of the number operators on∧F+ and

∧F−.

54


Of course, the generators of the free time evolution and of the global gauge transformationsare easy to compute because both operators are diagonal w.r.to the polarizationH = H+ ⊕ H−. In particular, we can set Γ(Ut)Ω = Ω,∀t ∈ R. However, the suggestedrelationship between Γ and dΓ is found to be true in the general setting:

Theorem 5.1.6 (Generators of Unitary Groups).Let A be a self-adjoint operator on H with [ε, A] Hilbert-Schmidt. Then dΓ(A) = : AΨ∗Ψ : asdefined in (5.1.24) is essentially self-adjoint on F and for its (unique) self-adjoint extensionon the Fock space F it is true that ei :AΨ∗Ψ: is an implementation of eiA.In particular, we can choose phases for the implementations such that

Γ(eiA) = ei dΓ(A) = ei :AΨ∗Ψ: (5.1.31)

Proof. [CaRu87] Proposition 2.1 ff.. This seems to be the first complete proof of this theorem.

Note that (5.1.31) fixes the phase of the implementation Γ(eiA) only in a neighborhood of theidentity where the exponential map is 1:1 from the Lie algebra on to the Lie group.

In physics, normal ordering is often introduced in an ad-hoc and merely formal way, simplyby “writing all the annihilators on the right of the creation operators”, so that it gives zero whenacting on the vacuum. This is why quite often there seems to be something mysterious or evendishonest about it. But in fact, the last theorem tells us that the second quantization in normalordering merely defines an action of the Lie algebra ures of Ures(H) on F . No big mystery there.

For A in the trace-class, AΨ∗Ψ and : AΨ∗Ψ : differ only by a constant multiple of theidentity. In fact, they just correspond to different “lifts” of A to the central extension ures

∼=ures ⊕ R that is acting on the Fock space. From the discussion in Chapter 3, we also knowthat as Γ defines only a projective representation of Ures(H), we are going to end up witha commutator anomaly in the representation of the corresponding Lie algebra in form of aLie algebra 2-cocycle. It should come as no surprise that we encounter the Schwinger-cocycle(4.1.10) again.

Proposition 5.1.7. (Schwinger Cocycle)For A,B ∈ ures we find

[dΓ(A),dΓ(B)] = dΓ([A,B]) + c(A,B)1 (5.1.32)

with the Schwinger cocycle c(A,B) = tr(A−+B+− −B−+A+−) computed in (4.1.10).

Proof. Because [dΓ(A),dΓ(B)] and dΓ([A,B]) are both second quantizations of [A,B], theydiffer only by a constant multiple cAB of the identity 7, i.e.

[dΓ(A),dΓ(B)] = dΓ([A,B]) + cAB · 1

Now dΓ([A,B]) is defined precisely in such a way that the vacuum expectation value 〈Ω,dΓ([A,B])Ω〉vanishes. Therefore,

cAB = 〈Ω, [dΓ(A),dΓ(B)]Ω〉7In the language of the previous chapters: [dΓ(A),dΓ(B)] and dΓ([A,B]) are just different lifts of

[A,B] ∈ ures to ures ∼= ures ⊕ C.

55


We take bases (fj)j∈N of H+ and (gk)k∈N of H− and compute:

〈Ω,dΓ(A)dΓ(B) Ω〉 = 〈Ω, AbaBa∗b∗ Ω〉

=∑j,k

∑l,m

(gk, Afj)(fl, Bgm)〈Ω, b(gk)a(fj)a∗(fl)b

∗(gm)Ω〉

=∑j,k

(gk, Afj)(fj , Bgk)〈Ω, b(gk)a(fj)a∗(fj)b

∗(gk)Ω〉

=∑j,k

(gk, Afj)(fj , Bgk) = tr(A−+B+−)

Similarly, 〈Ω,dΓ(B)dΓ(A)Ω〉 = tr(B−+A+−). And therefore

cAB = c(A,B) = tr(A−+B+− −B−+A+−).

Note that on ures ∩ I1(H), the proof simplifies considerably. Because then, we can use: AΨ∗Ψ : = AΨ∗Ψ− tr(A−−)1 and it is easy to check that [AΨ∗Ψ, BΨ∗Ψ] = [A,B]Ψ∗Ψ.Hence we find

[: AΨ∗Ψ : , : BΨ∗Ψ :] = [A,B]Ψ∗Ψ = : [A,B]Ψ∗Ψ : + tr([A,B]−−) · 1

and

tr([A.B]−−) = tr(A−+B+− +A−−B−− −B−+A+− −B−−A−−

)= tr

(A−+B+− −B−+A+−

)= c(A,B).

In Conclusion:We have sketched the quantization of the Dirac theory in the most established (rigorous) way.It should be noted that at no point in this construction did an infinite number of particlesappear and no reference to a Dirac sea or the like was made. But still, we have encounteredall the difficulties that we have hinted at in the introductory chapter and that were suggestedby the Dirac sea picture:

• Unitary transformations are not implementable as unitary operators on the Fock spaceunless they satisfy the Shale-Stinespring condition.

• Even for implementable operators, the implementation is well-defined only up to a com-plex phase.

• Second quantization of Hermitian operators has similar difficulties. Where it does exist,it gives rise to a commutator anomaly in form of the Schwinger term.

These results are the reason why a second quantization of the time evolution is, in general,impossible. The asymptotic case will turn out to be somewhat better behaved. But even then,at least the phase of the second quantized scattering operator S remains ill-defined.

56


5.1.2 C∗ - and CAR- Algebras

We take a quick look at Fock spaces and implementations of unitary transformations from theperspective of representation theory. This approach is very abstract but will reveal its beautyalong the way.

Definition 5.1.8 (C∗-Algebra).A unital Banach algebra (over C) is a complex, associative algebra B with 1, equipped with anorm ‖·‖ that makes it a complex Banach space and has the properties

i) ‖1‖ = 1 ii) ‖ab‖ ≤ ‖a‖ · ‖b‖, ∀a, b ∈ B

A C∗-algebra is a complex Banach algebra A together with an anti-linear involution∗ : A→ A satisfying:

iii) (a∗)∗ = a iv) (ab)∗ = b∗a∗ v) ‖a∗a‖ = ‖a‖2

∀a, b ∈ A. Properties ii)− v) together imply ‖a∗‖ = ‖a‖.

Definition 5.1.9 (C∗-homomorphism).A C∗-homomorphism is an algebra homomorphism h : A→ B between twoC∗-algebras A and B satisfying h(a∗) = h(a)∗, ∀a ∈ A.It can be shown that any such homomorphism is continuous with norm ≤ 1.

Examples 5.1.10 (C∗-algebra of bounded operators).Consider the Banach space B(H) of bounded operators on a Hilbert space H.On B(H) we have a ∗-involution T 7→ T ∗, where T ∗ denotes the adjoint of the boundedoperator T . From the well-known properties of Hermitian conjugation and of the operatornorm it follows that

(B(H), ‖·‖, ∗

)is a C∗-algebra. In fact, every C∗-algebra is isomorphic to

a sub-algebra of B(H) for suitable H.

Definition 5.1.11 (States of a C∗-algebra).A state of a C∗-algebra A is a complex-linear map ω : A→ C satisfyingω(1) = 1 and ω(a∗a) ≥ 0, ∀a ∈ A.

Examples 5.1.12 (Quantum States).For the C∗-algebra A = B(H) every Ψ ∈ H defines a state ωΨ by

ωΨ(T ) :=〈Ψ, T Ψ〉〈Ψ,Ψ〉

, ∀T ∈ B(H)

In the language of quantum theory, we would say that the state represented by the Hilbert-space vector Ψ ∈ H is characterized by all its “expectation values”.

Definition 5.1.13 (CAR-Algebra).A CAR-map into a C*-algebra A is an anti-linear map a : H → A satisfying the CAR-relations

a(f)a(g)∗ + a(g)a(f)∗ = a(f), a∗(g) =〈f, g〉 · 1 (5.1.33)

a(f)a(g) + a(g)a(f) = a(f), a(g) = 0 (5.1.34)

A CAR algebra over H is a C*-algebra A(H) together with a CAR-map a : H → A(H) havingthe following universal property: for every CAR-map a′ : H → B into a C*-algebra B there is

57


a unique C*-homomorphism h : A(H)→ B s.t. a′ = h a.

H a //

a′

A(H)

h||B

(5.1.35)

Such a CAR-algebra can be explicitly constructed and by the universal property it is uniqueup to canonical isomorphism.

It is probably more intuitive to think of the CAR Algebra as the free associative algebragenerated by H modulo the canonical anti-commutation relations (5.1.10). Mathematicians,however, like to define it by the universal property as above which is rather abstract, butvery elegant to work with. We might get a hint of how powerful the universal property is,by clarifying the relationship between field operators as introduced in the last section andrepresentations of the abstract CAR-algebra defined above. Recall that a field operator is amap Ψ : H → B(F) from the Hilbert-space into the C∗-algebra of bounded operators on theFock-Space, satisfying the CAR (5.1.10). By the universal property of the CAR-algebra thereexists a C*- homomorphism π : A(H) → B(F) s.t. Ψ = π a. This homomorphism π is thena representation of the CAR-algebra A(H) on F .

Proposition 5.1.14 (Fock Representation).

The field operator Ψ defined in (5.1.9) induces an irreducible representation of the CAR-algebraof H on the Fock space

F = F(H+,H−) :=∧H+ ⊗

∧H−, (5.1.36)

called the Fock representation. (Here we take, for simplicity, C to be the anti-unitary map ontothe complex conjugated space).

Similarly, we can define for an arbitrary polarization W ∈ Pol(H) the field operator ΨW (f) =

a(PW f) + b∗(PW⊥f) and call the induced representation on F(W,W⊥) :=∧W ⊗

∧W⊥ the

Fock representation w.r.to the polarization H = W ⊕W⊥.

5.1.3 Representations of the CAR Algebra

Mathematically, a “Fock space” can be understood as a Hilbert space, arising as the repre-sentation space of an irreducible representation of an abstract CAR-algebra (in the fermioniccase), or a CCR-algebra (in the bosonic case). There are various ways to get irreduciblerepresentations of a CAR-algebra and they are generally not equivalent to each other.

Definition 5.1.15 (Equivalent Representations).Consider the CAR algebra A(H) ober H. Two representations π and π′ of A(H) on Fockspaces F and F ′ are equivalent if there exists a unitary transformation T : G → G′ withT π(a(f)) = π′(a(f))T, ∀f ∈ H.

We generalize the notion of “implementability” introduced in Def. 5.1.3 to arbitrary represen-tations of the CAR-algebra.

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Definition 5.1.16 (Implementation of Unitary Transformation).Let U ∈ U(H) a unitary operator. If a : H → A(H) is a CAR-map, so is aU . By the universalproperty of the CAR-algebra, there exists a unique C∗-homomorphism βU : A(H) → A(H)

with βU a = a U . βU is the Bogoljubov transformation corresponding to U .

Let π : A(H)→ F be a representation of the CAR-algebra on the Fock space F .U (or βU ) is called implementable on F if there exists a unitary operator U on F such thatUπ(a(f)) = π(a(Uf))U , ∀f ∈ H. U is then called an implementation of U on F .

In other words: U is implementable on the Fock space F if and only if the two CAR-representations π and πβU are unitarily equivalent. Therefore, the “implementation problem”and the “equivalence problem” are closely related.

GNS-representations

We have explicitly constructed the Fock representation on F =∧H+ ⊗

∧H−, by defining

creation and annihilation operators and finally the field operator Ψ (5.1.9). This, however,was just a very special example of a Fock space and a representation of the CAR-algebra. Theseminal paper of Shale and Stinespring is actually formulated for spin-representations of theinfinite-dimensional Clifford algebra of H which is closely related to the CAR-algebra A(H)

([SS65], see [G-BVa94] for a more detailed construction). One of the most important methodsfor obtaining representations of a CAR algebra are the so-called GNS constructions (after I.M.Gelfand, M.A. Neumark and I.E. Siegel.) Given a state ω on A(H), the GNS-constructionyields a Hilbert space Hω, a representation πω on Hω and a unit vector Ωω ∈ Hω (the “GNS-vacuum”), such that

ω(a) = 〈Ωω, πω(a) Ωω〉 , ∀a ∈ A(H) (5.1.37)

and such that the action of πω(A(H)) on Ωω generates a dense subset of Hω.

In particular, given an orthogonal projection Q, i.e. a self-adjoint operator on H with Q2 = Q,we can define a state ωQ by setting ωQ(1) = 1 and fixing the two-point functions

ωQ(a∗(f)a(g)) := 〈g,Qf〉H (5.1.38)

for f, g ∈ H and a, the CAR-map defining A(H). Then, the corresponding GNS-construction(FQ, πQ,ΩQ) satisfies

〈ΩQ, πV (a∗(g)a(f))ΩQ〉 = 〈g,Qf〉H = 〈Qf,Qg〉H (5.1.39)

and, using a(g)a∗(f) = 〈g, f〉H1− a∗(f)a(g),

〈ΩQ, πQ(a(g)a∗(f))ΩQ〉 = 〈f, g〉H − 〈Qf,Qg〉H = 〈(1−Q)f, (1−Q)g〉H. (5.1.40)

In particular, we read off

πQ(a(g))ΩQ = 0, for g ∈ ker(Q)

πQ(a∗(f))ΩQ = 0, for f ∈ im(Q)(5.1.41)

so that πQ a : H → B(FQ) acts like a field operator on the vacuum ΩQ.Indeed, for W ∈ Pol(H) and Q = P⊥W , the two-point functions of the GNS-construction are

59


the same as the two-point functions of the field operator ΨW on∧PWH⊕

∧PW⊥H, because

〈Ω,Ψ∗(f)Ψ(g)Ω〉 =⟨CPW⊥f, CPW⊥g

⟩=⟨g, PW⊥f

⟩H = 〈g,Qf〉H .

This implies that the two representations are actually equivalent.Now, in the language of representation theory we can formulate another, very general versionof the Shale-Stinespring theorem, also known as the theorem of Powers and Størmer.

Theorem 5.1.17 (Powers, Størmer 1969).The GNS-representations (FP , πP ,ΩP ) and (FQ, πQ,ΩQ) corresponding to orthogonal projec-tions P and Q on H are unitary equivalent if and only if P −Q is a Hilbert-Schmidt.

Proof. [PoSt70]

Applying this result to the “implementation problem” we find ones more that U ∈ U(H) isimplementable on F =

∧H+ ⊗

∧CH− if and only if P− − UP−U∗ is Hilbert-Schmidt i.e. if

and only if U+− and U−+ are Hilbert-Schmidt i.e. if and only if U ∈ Ures(H).

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5.2. THE INFINITE WEDGE SPACE CONSTRUCTION

5.2 The Infinite Wedge Space Construction

In the light of realization that the unitary time evolution cannot be implemented on a fixedFock space, my esteemed teachers and colleagues Dirk Deckert, Detlef Dürr, Franz Merkl andMartin Schottenloher developed a formulation of the external field problem in QED on time-varying Fock spaces. The Fock spaces are realized as “infinite wedge spaces” constructed from“Dirac seas” over a chosen polarization class. This construction is nice for several reasons:

1. It is very “down to earth” and close to the physical intuition.

2. It is designed to highlight all the ambiguities and choices involved in the construction,rather than hiding them.

3. The formalism is very flexible and well-suited for the treatment of different polarizationclasses and the implementation of the time evolution on varying Fock spaces.

Of course, Deckert et.al. are consistently ignoring the geometric structure underlying the con-struction (as we will see). However, in doing so, it becomes clear that this geometric structure,as beautiful as it might be, is not essential for the understanding of the crucial (physical) prob-lems. Whether it is helpful in finding a solution is a different question that will be discussedin subsequent chapters.

The basic idea of the infinite wedge space construction is this:

• For a fixed equal charge polarization class, we consider certain maps from an index-space` into the Hilbert space H, with image in the respective polarization class. We call thesemaps seas. Intuitively, they will represent states of the Dirac sea, i.e. states of infinitelymany fermions.

• If we fix a basis (ek)k∈N in `, we can think of a such a sea Φ : ` → H as the infiniteexterior product Φ(e0) ∧ Φ(e1) ∧ Φ(e2) ∧ . . .

• We generalize the finite-dimensional scalar product (1.2.1) (also known as Slater deter-minant) by the infinite-dimensional Fredholm determinant i.e. by

〈Φ,Ψ〉 = det(Φ∗Ψ) = limN→∞

det(〈Φ(ei),Ψ(ej)〉

)i,j≤N . (5.2.1)

For this to be well-defined, we have to impose further restrictions on the permissibleseas. This leads us to an equivalence relation, defining classes of seas.

• Finally, we get a linear space by taking the algebraic dual of such a sea class and formingthe completion with respect to before mentioned scalar product. This will be the Fockspace realized as an infinite wedge space over the chosen Dirac sea class.

For the remainder of this chapter, let ` be an infinite-dimensional, separable, complex Hilbert-space. ` will serve as an index space. A convenient choice is therefore ` = `2(C).

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Definition 5.2.1 (Dirac Seas).

Let Seas(H) be the set of all bounded, linear maps Φ : `→ H such that im(Φ) ∈ Pol(H) andΦ∗Φ : `→ ` has a determinant, i.e. Φ∗Φ ∈ Id+ I1(`).

Let Seas⊥(H) denote the subset of all Φ ∈ Seas(H) that are also isometries.

For any equal-charge polarization class C ∈ Pol(H)/ ≈0 let Ocean(C) be the set of all Φ ∈Seas⊥(H) with im(Φ) ∈ C.

Definition 5.2.2 (Dirac Sea Classes).For Φ,Ψ ∈ Seas(H) we introduce the relation

Φ ∼ Ψ ⇐⇒ Φ∗Ψ ∈ Id+ I1(`)

This defines an equivalence relation on Seas(H) ([DeDuMeScho], Cor. II.9) . The equivalenceclasses are called Dirac sea classes and denoted by S(Φ) ∈ Seas(H)/ ∼.

Lemma 5.2.3 (Connection between ∼ and ≈0).Given C ∈ Pol(H)/≈0 and Φ ∈ Ocean(C) we find

C = im(Ψ) | Ψ ∈ Seas⊥(H) such that Ψ ∼ Φ.

In other words: Ψ ∼ Φ in Seas⊥(H)⇒ im(Ψ) ≈0 im(Φ) in Pol(H)

and W ∈ C = [im(Φ)]≈0 ⇒ ∃Ψ ∈ Seas⊥(H) : Ψ ∼ Φ ∧ im(Ψ) = W .

Proof. See Lemma II.12 in [DeDuMeScho] or note that the statement follows as a reformula-tion of our Lemma 5.3.2 (“Existence of admissible basis”) and the remark following Definition5.3.1(“Admissible Basis”). A polar decomposition can be used to make the Dirac sea/admissiblebasis isometric.

Construction 5.2.4 (Formal Linear Combinations).

1. For any set S, let C(S) denote the set of all maps α : S → C for which the supportΦ ∈ S | α(Φ) 6= 0 is finite. For Φ ∈ S, let [Φ] ∈ C(S) denote the algebraic dual, i.e.the map satisfying [Φ](Φ) = 1 and [Φ](Ψ) = 0 for Φ 6= Ψ ∈ S. Thus, C(S) consists ofall finite formal linear combinations α =

∑Ψ∈S α(Ψ)[Ψ] of elements of S with complex

coefficients.

2. Now, let S ∈ Seas(H)/∼ be a Dirac sea class.We define the map 〈·, ·〉 : S × S → C, (Φ,Ψ) 7→ 〈Φ,Ψ〉 := det(Φ∗Ψ).The determinant exists and is finite by definition of ∼.

3. For S ∈ Seas(H)/∼, let 〈·, ·〉 : C(S) × C(S) → C denote the sesquilinear extension of〈·, ·〉 : S × S → C to the linear space C(S) defined as follows:For α, β ∈ C(S),

〈α, β〉 :=∑Φ∈S

∑Ψ∈S

α(Φ)β(Ψ) det(Φ∗Ψ). (5.2.2)

The bar denotes the complex conjugate. Note that the sums consist of at most finitelymany nonzero summands. In particular, we have 〈[Φ], [Ψ]〉 = 〈Φ,Ψ〉 for Φ,Ψ ∈ S.The sesquilinear form 〈·, ·〉 on C(S) is Hermitian and positive-semidefinite8.Therefore, it defines a semi-norm on C(S) by ‖α‖ :=

√〈α, α〉.

8see[DeDuMeScho], Lemma II.14 or our Proposition 5.3.12 “Hermitian Form”.

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Definition 5.2.5 (Infinite Wedge Space).Let FS be the completion of C(S) with respect to the semi-norm ‖·‖.FS is an infinite-dimensional, separable, complex Hilbert space.We will refer to it as the infinite wedge space over the Dirac sea class S.

By∧

: S → FS we denote the canonical map∧

Φ := [Φ] coming from the inclusions

S → CS → FS .

Note that the null-space NS := α ∈ C(S) | ‖α‖ = 0 is factored out in the process ofcompletion.

Construction 5.2.6 (The Left Operation).

U(H) acts on Seas(H) from the left by (U,Φ)→ UΦ.This extends to a well-defined map

S U−→ US := UΦ | Φ ∈ S

between Dirac sea classes.

For U ∈ U(H), the induced left operation LU : C(S) → C(US), given by

LU

(∑Φ∈S

α(Φ)[Φ]

)=∑Φ∈S

α(Φ)[UΦ],

is an isometry with respect to the Hermitian forms 〈·, ·〉 on C(S) and C(US).Consequently, it extends to a unitary map LU : FS → FUS between infinite wedge spaces,characterized by LU (

∧Φ) =

∧(UΦ).

This extends immediately to unitary maps between different Hilbert spaces H and H′.

Construction 5.2.7 (The Right Operation).

Let Gl−(`) := R ∈ Gl(`) | R∗R ∈ Id+ I1(`).Gl−(`) acts on Seas(H) from the right by (Φ, R)→ ΦR. This extends to a well-defined map

S R−→ SR := ΦR | Φ ∈ S

between Dirac sea classes. For S ∈ Seas(H)/∼ and R ∈ GL−(`) we have an induced operationfrom the right, namely RR : C(S) → C(SR) given by

RR

(∑Φ∈S

α(Φ)[Φ]

)=∑Φ∈S

α(Φ)[ΦR].

This map is angle-preserving w.r.to the Hermitian forms, i.e. an isometry up to scaling, since

det((ΦR)∗(ΨR)

)= det(R∗ΦΨR) = det(R∗R) det(Φ∗Ψ)

∀R ∈ Gl−(`) and Φ,Ψ ∈ S and therefore, for all α, β ∈ C(S),

〈RRα,RRβ〉 = det(R∗R) 〈α, β〉 .

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In particular, we have RR[NS ] ⊆ NSR. It follows that for every R ∈ Gl−(`), the operationRR : C(S) → C(SR) induces a bounded linear map RR : FS → FSR between the infinite wedgespaces, characterized by RR(ΛΦ) = Λ(ΦR) for Φ ∈ S. This map is unitary, up to scaling.The definition extends immediately to maps between different index spaces ` and `′.

We can think of the right action as basis transformations on the Hilbert space or, with asomewhat greater leap of imagination, as “rotations” of the Dirac seas. The seas will stay inthe same Dirac sea class if and only if the transformations are “small” in the sense that theyvary from the identity only by a trace-class operator or, in other words, have a determinant.Pictorially speaking, we “rotate” a Dirac sea just “a little”, if we don’t stir too much deep downin the sea. This intuition is made precise in the following Lemma:

Lemma 5.2.8 (Uniqueness up to a Phase).Let R ∈ Gl−(`) and S ∈ Seas(H)/∼ a Dirac sea class. Then, SR = S if and only if R has adeterminant, i.e. iff R ∈ GL1(`). In this case, the right operation RR : FS → FS correspondsto multiplication with det(R) on FS . In particular, if R ∈ U(`) ∩ GL1(`), the right operationRR corresponds to multiplication by a complex phase ∈ U(1).

Since the right-action is a well-defined map between Dirac sea classes, it follows immediatelyfor another operator Q ∈ GL−(`) that SR = SQ if and only if Q−1R has a determinant. Inthis case, RR = det(Q−1R)RQ on FS .

Proof. Let S ∈ Seas(H)/∼. For any Φ ∈ S, Φ∗(ΦR) has a determinant if and only if R has adeterminant, since Φ∗Φ ∈ Id+ I1(`) by definition. Thus:

Φ ∼ ΦR ⇐⇒ S = SR ⇐⇒ R ∈ Gl1(`)

Now, on C(S) with the semi-norm defined by 〈·, ·〉:

‖[ΦR]− (detR)[Φ]‖2 = det((ΦR)∗(ΦR))− (detR) det((ΦR)∗Φ)

−detR det(Φ∗ΦR) + |detR|2 det(Φ∗Φ)

= 2|detR|2 det(Φ∗Φ)− 2|detR|2 det(Φ∗Φ) = 0. (5.2.3)

Therefore, RR = det(R) · Id on FS .

Recall that the construction of the Fock space as an infinite wedge space involves twoconsecutive choices. The first one (usually determined by the physics) is the choice of apolarization class C ∈ Pol(H). Afterwards, we have a more or less arbitrary choice of apolarization class S ∈ Ocean(C)/∼. As this is an equivalence class, it is uniquely determinedby a “reference polarization” Φ ∈ Ocean(C). This duality is reflected in the duality of left-and right- operations. The operations from the left are transformations between polarizationclasses. The unitary transformation induced by U ∈ U(H) stays in the same polarization classif and only if it satisfies the Shale-Stinespring condition (1.2.10). It preserves the charge, if andonly if the (++)-component w.r.t the corresponding polarization has index 0. The operationsfrom the right, on the other hand, are transformations between different Dirac sea classes withinthe same ocean i.e. between seas with image in the same polarization class. We see how themathematical structure at hand gives us a very natural way to handle unitary transformationsbetween Fock spaces.

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The left-action alone would correspond to the product-wise lift

Φ(e0) ∧ Φ(e1) ∧ Φ(e2) ∧ · · · LU−−−−→ UΦ(e0) ∧ UΦ(e1) ∧ UΦ(e2) ∧ · · ·

But this alone is not very helpful, in general. In addition, we need a suitable right-operationRR to rotate the seas back into the desired Dirac sea class. If U preserves an equal-chargepolarization class C ∈ Pol(H)/ ≈0, i.e. U ∈ U0

res(H, C;H, C) a right-operation can be cho-sen in such a way as to make RRLU a unitary transformation on the Fock space FS for afixed S ∈ Ocean(C)/∼. If, however, U maps one polarization class C into a different one, i.e.U ∈ U0

res(H, C;H, C ′) with C ′ 6= C, the best we can do is to implement U as a unitary transfor-mation between different Fock spaces FS and FS′ for S ∈ Ocean(C)/∼ and S ′ ∈ Ocean(C ′)/∼.The right-operation then has to be such that USR = S ′.

The previous Lemma tells us that the implementations are determined only up to a com-plex phase. This means that the group that acts on the Fock space contains an additionalU(1)-freedom besides the information contained in U0

res(H, C;H, C) = Ures(H). Indeed, itcorresponds to the central extension Ures(H) of Ures(H) by U(1) studied in Chapter 4 (cf.[Cor.5.4.3]). These considerations lie at the heart of the following crucial theorem.

Theorem 5.2.9 (Lift of Unitary Transformations, [DeDuMeScho] Thm. II.26).For given polarization classes C,C ′ ∈ Pol(H)/≈0 let S ∈ Ocean(C)/∼ and S ′ ∈ Ocean(C ′)/∼.Then, for any unitary map U : H → H, the following are equivalent:

i) U ∈ U0res(H, C;H, C ′).

ii) There is R ∈ U(`) such that USR = S ′, and hence RRLU maps FS to FS′ .In this case, if R′ ∈ U(`) is another map with USR′ = S ′, thenRR′LU = det(R′R∗)RRLU with det(R′R∗) ∈ U(1).

Corollary 5.2.10 (U(`) acts transitively on oceans).Setting U = 1H in the last theorem it follows immediately that

Ocean(C)/∼ = SR | R ∈ U(`).

for any given C ∈ Pol(H)/≈0 and S ∈ Ocean(C)/∼.

In other words: U(`) acts transitively on Ocean(C)/∼ from the right.

Corollary 5.2.11 (Equivalence of Infinite Wedge Spaces).Let S,S ′ ∈ Ocean(C)/∼ two Dirac sea classes over the same polarization class C ∈ Pol(H)/≈0.There exists R ∈ U(`) such that RR : FS → FS′ is an isomorphism.If R′ is another map providing such an isomorphism, then R∗R′ ∈ U1(`) has a determinantand RR′ = det(R′R∗)RR. In particular, FS = FS′ if and only if R ∈ U1(`).

Theorem 5.2.9 is a generalization of the Shale-Stinespring theorem in the language of theinfinite wedge spaces. Setting C = C ′ = [H+] it reproduces the well-known result that aunitary operator U ∈ U(H) is implementable on a fixed Fock space F over the polarizationclass [H+] if and only if U ∈ Ures(H). Together with Lemma 5.2.8, the theorem implies thatthe implementation of a unitary operator (on a fixed Fock space or as a map between differentFock spaces) is unique up to a complex phase. So again, we have encountered the infamousgeometric phase of QED.

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5.3. THE GEOMETRIC CONSTRUCTION

5.3 The Geometric Construction

In this section, we are going to follow the construction of the fermionic Fock space as presented,for example, in [PreSe] or [Mi]. The Fock space will be constructed from holomorphic sections inthe dual of the determinant line-bundle over the complex (restricted) Grassmannian manifold.We will refer to this as the geometric construction of the Fock space. It as quite an elegantapproach for several reasons:

1. It reveals and exploits a remarkably elegant geometric structure that appears very nat-urally in our mathematical framework.

2. Polarizations, seas and their basis transformations are embraced by the geometric de-scription, were they appear as base-manifold, fibre bundle and structure group. Thisprovides an elegant and concise formalism, illustrating the relations between the objectsinvolved.

3. The central extensions GLres(H) of GLres(H) and Ures(H) of Ures(H) constructed inChapter 4 act most naturally on this Fock space.

We will follow the pertinent literature in constructing the Fock space for the standard po-larization H = H+ ⊕ H−. This will serve as an exemplary illustration of the general case.The construction can be easily applied to arbitrary polarizations (and polarization classes),matching the generality of the infinite wedge space construction. This is outlined at the endof the section. We are also conforming to the mathematical literature in using H+ insteadof H− for the polarization that – intuitively – plays the role of the “Dirac sea” in the geo-metric construction of the Fock space. So the reader should note that the convention used inthis section is in conflict with the physical convention used in §5.1, as well as in chapters 7 and 8.

We have already argued that polarizations, which are points on the infinite-dimensionalGrassmanian manifold Gr(H), can be thought of as “projective states” of infinitely manyfermions. Of course, we are not satisfied with a projective description, but aiming for afull-blown Hilbert space structure. In particular, we want to generalize the scalar product

〈v1 ∧ · · · ∧ vn, w1 ∧ · · · ∧ wn〉 = det(〈vi, wj〉)i,j

from finite exterior products to infinite particle states. However, as we have seen before, whilethis expression is always well-defined in the finite-dimensional case, the infinite-dimensionallimes exists only if we impose further compatibility-conditions on the choices of Hilbert-basesfor different polarizations. In the infinite wedge space construction, this was done by identifying“Dirac sea classes”, on which the Hermitian form is well-defined. In the geometric construction,the corresponding concept is that of admissible bases.

Definition 5.3.1 (Admissible Basis).Let W ∈ Gr0(H). An admissible basis for W is a bounded isomorphism w : H+ →W with theproperty that P+ w has a determinant.

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It might be helpful to note:

• W = im(w) ∈ Gr(H) iff P+w is Fredholm and P−w Hilbert-Schmidt.

• W = im(w) ∈ Gr0(H) iff in addition ind(P+w) = 0.

• w is admissible basis of W ∈ Gr0(H) iff in addition P+w ∈ Id+ I1(H+).

Lemma 5.3.2 (Existence of Admissible Bases).

Every W ∈ Gr0(H) has an admissible basis.

Proof. Let W ∈ Gr0(H). Set w := PW |H+: H+ → W . P+w has a determinant, because

1H+−P+w =(P+−P+PWP+

)∣∣H+

and P+−P+PWP+ = P+PW⊥P+ = (PW⊥P+)∗(PW⊥P+) ∈I1(H) (Lemma 2.1.3). We also know that w is a Fredholm operator of index 0(bc. charge(W,H+) = ind(PW |H+) = 0 ), which means that the kernel of w and its cokernelin W are of equal and finite dimension. Therefore, we can define w : H+ → W by settingw = w on ker(w)⊥ and extending it to the whole H+ in such a way that it maps ker(w) to acomplement of im(w) in W . Thus, by construction, im(w) = W and P+ w has a determinantas it differs from P+w by a finite-rank operator only. Using a polar decomposition, we can alsomake w unitary.

Lemma 5.3.3 (Relationship between Admissible Bases).Let w : H+ → W be an admissible basis for W ∈ Gr(H)(H). Then, w′ : H+ → W is anotheradmissible basis for W , if and only if w′ = w L for some L ∈ GL1(H+).

Proof. If w′ = w L, then w′ is an isomorphism from H+ to W and P+w′ = (P+w)L has a

determinant because P+w and L do. Therefore, w′ is an admissible basis for W .Conversely, if w′, w are two admissible bases for W , we set L := w−1 w′ : H+ → H+.Clearly, L is invertible. Furthermore, P+wL = P+w

′ ∈ Id+ I1(H+) and alsoP+w ∈ Id+ I1(H+), which implies L ∈ Id+ I1(H+).

Arbitrary charges

Recall that the Grassmanian Gr(H) has Z connected components, corresponding to the rela-tive charges of the polarizations (w.r.to H+). Usually we are only interested in the connectedcomponent Gr0(H) of the initial vacuum ≈ H+. The geometric construction over Gr0(H) willyield the zero-charge sector of the fermionic Fock space. Note that the infinite wedge spaceconstruction also yields Fock spaces of “constant charge”. We can easily extend the constructionto a “full” Fock space by taking the direct (orthogonal) sum of infinite wedge spaces over polar-ization classes with different relative charges. However, such a construction is not canonical,as it involves a choice of a Dirac sea class for every such charge sector. Thus it should come asno surprise that the same is true for the geometric construction as well. It inherits arbitrarycharges very naturally, however, the construction is not canonical. It involves a choice of a(polarized) basis.

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For the remainder of this section we fix an orthonormal basis (ek)k∈Z of H such that (ek)k≤0

is an ONB of H− and (ek)k≥0 is an ONB of H+. For any n ∈ Z we write

H≥−n := span(ek | k ≥ −n).

By P (n)+ we denote the orthogonal projection PH≥−n onto H≥−n

From the construction of the complete GLres(H) in Section 4.1.2, we recall the shift-operatorσ defined by σ(ek) := ek+1. For any d, n ∈ Z, σd maps H≥−n to H≥−n+d.9

Definition 5.3.4 (Admissible Basis for arbitrary Charges).Let W ∈ Gr(H) with charge(W,H+) = n, i.e. W lies in the connected component Grn(H) ofGr(H). An admissible basis for W ∈ Grn(H) is a bounded linear map w : H≥−n → H withim(w) = W s.t. w : H≥−n →W is an isomorphism and P (n)

+ w has a determinant.

We denote the set of all admissible bases by St(H) and the subset of admissible bases forGrn(H), n ∈ Z by St(n)(H). On St(n)(H), we let GL1(H+) act from the right by

wL7−→ w σ−n L−1 σn

With a little abuse of notation, we can just write wL−1 for the right action and P+ instead ofP

(n)+ for the orthogonal projection whenever the specific index is clear or irrelevant. With this

sneaky notation, we can forget about the different charges most of the time. The constructionwill look the same over any connected component.

Definition 5.3.5 (Stiefel Manifold).The set St(H) of all admissible bases for the polarizations in Gr(H) carries the structure of aninfinite-dimensional manifold, called the (restricted) Stiefel manifold St(H). The topology isgiven by the metric

d(w,w′) = ‖P+(w − w′)‖1 + ‖P−(w − w′)‖2. (5.3.1)

The Stiefel manifold has Z-connected components, corresponding to the connected componentsof Gr(H). Actually, it follows from the previous discussion that π : St(H) → Gr(H) withπ(w) := im(W ) is naturally a principle GL1(H+)-bundle over Gr(H). The fibre π−1(W ) overa basepoint W ∈ Gr(H) is just the set of admissible bases for W and GL1(H+) acts on thesefibers from the right as described above.

Proposition 5.3.6 (det is holomorphic).The (Fredholm-) determinant det : GL1(H+)→ C\0 defines a one-dimensional, holomorphicrepresentation of the Lie group GL1(H+).

Proof. Recall that the Lie group structure is given by the embedding

GL1(H+) → I1(H+); 1 +A 7→ A

of GL1(H+) into the trace-class. Now note that it suffices to show complex differentiability inthe identity, because for (1+A), (1+B) ∈ GL1(H+), we have by the multiplication-rule

det(1 +A)− det(1 +B) = det(1+B)(det((1+A)(1+B)−1)− det(1)

)9If we seek more generality, we can choose for each n ∈ Z an arbitrary subspace Wn ∈ Gr(H) with

charge(Wn,H+) = n, and a one-parameter group of unitary operators translating between them.

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with (1 + A)(1 + B)−1 = (1 + C) ∈ GL1(H+). And since (1 + C)(1 + B) = (1 + A) ⇐⇒1+ C +B + CB = 1+A ⇐⇒ C(1+B) = A−B, we find C → 0 ⇐⇒ B → A in I1(H+).Now, by definition:

det(1+A) :=

∞∑k=0

tr(∧k

(A)).

If (λn)n are the eigenvalues of A (A is a compact operator), tr(∧k

(A))

=∑

i1<···<ikλi1 · · ·λik .

Using a polar decomposition, we can write A = U |A|, with a partial isometry U , and∧k

(A) =∧k(U)

∧k(|A|). The eigenvalues (µn)n of the positive operator |A| are the singular values of

A. We derive: ∥∥∧k(A)∥∥

1= tr

(∧k(|A|)

)=

∑i1<···<ik

µi1 · · ·µik

≤ 1

k!

∑i1,...,ik

µi1 · · ·µik =1

k!‖A‖k1 .

Therefore, we conclude:

|det(1+A)− 1− tr(A)| =∣∣ ∞∑k=2

tr(∧

k(A))∣∣ ≤ ∞∑

k=2

‖A‖k1k!≤ exp(‖A‖1).

This means that det is continuously differentiable in 1 with det′(1)X = tr(X) and henceholomorphic on the complex Banach Lie group GL1(H+).

Definition 5.3.7 (Determinant Line Bundle).We have a principle-GL1(H+)-bundle St(H) over Gr(H) and a one-dimensional, holomorphicrepresentation of GL1(H+), given by det : GL1(H+) → C. We define the determinant bundleas the corresponding associated line bundle

DET = DET(St(H),Gr(H),det

).

As St(H) has infinitely-many connected components, indexed by Z, so does DET . The con-nected component over St(n)(H) will be denoted by DETn.

In less technical terms, “associated line-bundle” means the following: Consider the set St(H)×Cwritten as

(W,w, λ) | λ ∈ C,W ∈ Gr(H), w admissible basis for W .

We introduce the equivalence relation

[W,w, λ] ∼ [W ′, w′, λ] :⇐⇒ W ′ = W, w′ = w L and λ′ = det(L)−1λ

i.e. [W,w,det(L)λ] ∼ [W,w L, λ], for L ∈ GL1(H+). (5.3.2)

The set of equivalence classes [W,w, λ], together with the projectionπ : [W,w, λ]→W = im(w) (induced by that in St(H)) defines a holomorphic line bundle(

St× C)/GL1 =: DET

π−−→ Gr(H).

Further on, we will usually drop the first entry and write [w, λ] for [im(w), w, λ].

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We might get a better intuition for this construction by thinking about the fibre overW ∈ Gr(H) in DET as the one-dimensional complex space spanned by a formal expression

π−1(W ) 3 [W,w, 1] = [w, 1] ' w0 ∧ w1 ∧ w2 ∧ w3 ∧ . . . (5.3.3)

where wjj≥0 is a basis of W . So, intuitively, the determinant bundle over Gr(H) containsprecisely the information that we expect to be encoded in the fermionic Fock space.However, so far we don’t even have a linear structure on the “state-space”, except of course forthe C1-structure on the individual fibres. But there’s no meaningful way of “adding” pointsfrom different fibers of the bundle. Therefore, the idea is to consider sections of the determinantline bundle which do naturally form an (infinite-dimensional) complex vector space Γ(DET ).Which section would then correspond to a state of the form (5.3.3) though? In analogy to theinfinite wedge space construction, where we got a linear space by taking the algebraic dual ofa Dirac sea class, we might think of the section that picks out the point [w, 1] over W ∈ Gr(H)

and is zero everywhere else. But such a section is not even continuous and wouldn’t do justiceto the beautiful geometric structure we have so far.

We can however take sections in the dual bundle DET ∗ and identify w ∈ DET with thesection ξw ∈ Γ(DET ∗) defined by ξw([z, λ]) := λ det(w∗z). Such a section is not only contin-uous, it is even holomorphic with respect to the holomorphic structure induced by Gr(H) (seealso [PreSe], §10).

Remark 5.3.8 (Holomorphic sections).We denote by DET ∗ the dual bundle of the determinant-bundle DET and by Γ(DET ∗) thespace of holomorphic sections in DET ∗. A holomorphic section Ψ of DET ∗ is a holomorphicmap DET → C which is linear in every fibre. This corresponds to a holomorphic map

ψ : St→ C with ψ(z L) = det(L) · ψ(z), ∀L ∈ GL1(H+), (5.3.4)

since then and only then is Ψ([z, λ]) := λ · ψ(z) a well defined map DET → C, as we see fromΨ([z L, λ/det(L)]) = λ

det(L) ψ(z L) = λψ(z) = Ψ([z, λ]).

Similarly, a holomorphic section Φ of DET would correspond to a holomorphic map

φ : St→ C with φ(z L) = det(L)−1 φ(z) , ∀L ∈ GL1(H+). (5.3.5)

Due to the factor of det(L)−1 on the right-hand-side, points arbitrarily close in St(H) can bemapped to points arbitrarily far in C, which contradicts the existence of a bounded derivative.Hence DET doesn’t allow any holomorphic sections, except for the zero section.

Construction 5.3.9 (Embedding of DET in Γ(DET ∗)).Consider the map Φ : St× St→ C,

Φ(z, w) =

det(z∗w) ; for ind(P+z) = ind(P+w)

0 ; else.

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This is well defined, since for ind(P+z) = ind(P+w) = n,

z∗w = z∗P(n)+ w + z∗P

(n)− w =

(P(n)+ z)∗︸︷︷︸

∈Id+I1(H≥−n)

(P(n)+ w)︸︷︷︸

∈Id+I1(H≥−n)

+ (P(n)− z)∗︸︷︷︸

∈I2(H<−n,H≥−n)

(P(n)− w)︸︷︷︸

∈I2(H≥−n,H<−n)

∈ Id+ I1(H≥−n).

Now, for any fixed z ∈ St(H), the map

w 7−→ Φ(z, w) =: ξz(w)

is holomorphic with Φ(z, w L) = det(z∗wL) = det(z∗w) det(L) for L ∈ GL1(H+) and hencedescends to a holomorphic section of DET ∗, that will be denoted by the same symbol ξz ∈Γ(DET ∗). This defines an anti-linear map ξ : DET → Γ(DET ∗) by

ξ([z, λ]) := λξz = λΦ(z, ·). (5.3.6)

Note thatξzL = det(L) ξz, ∀L ∈ GL1(H+) (5.3.7)

thus ξ has to be anti-linear, because [z L, λ] = [z,det(L)λ] in DET .

Deleting the zero-section from DET , we get an injection ξ : DET× → Γ(DET ∗).

This construction contains almost everything we need. We will construct the Fock space fromthe space of holomorphic sections of DET ∗, with the Hermitian scalar product given by thedeterminant. But first, we need good coordinates to handle this.

Definition 5.3.10 (Pflücker Coordinates).Let (ej)j∈Z be the ONB of H chosen above.

i) We denote by S the set of increasing sequences S = (i0, i1, i2.i3, . . . ) ∈ ZN0

s.t. ik+1 = ik + 1 for large enough k, i.e. the sequences S ∈ S contain only finitely manynegative indices and all but finitely many positive indices.

ii) For S ∈ S, we define the charge c(S) to be the unique number c ∈ Z with ik = k − c forall k large enough.

iii) For S = (i0, i1, i2.i3, . . . ) ∈ S with c(S) = n we define

HS := span(eik | ik ∈ S

)= span(ei0 , ei1 , ei2 , . . . )

andwS : H≥−n → H , wS(ek) := ei(k+n)

.

This is an admissible basis for HS since by construction, it differs from the identity onH≥−n by a finite rank matrix only.

iv) We denote by ΨS the section ξwS ∈ Γ(DET ∗), for S ∈ S.

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Examples 5.3.11 (Pflücker coordinates).

• S0 = N = (0, 1, 2, 3, . . . )⇒ c(S0) = 0. We call the corresponding state

ξwS0:= Ψ0 ≈ e0 ∧ e1 ∧ e2 ∧ e3 . . .

the free vacuum state.

• S1 = (−2,−1, 0, 1, 2, . . . )⇒ c(S1) = 2.This corresponds to the state e−2 ∧ e−1 ∧ e0 ∧ e1 ∧ . . . where 2 negative energy states(e−2 and e−1) are occupied and all positive energy states ej≥0 are occupied.

• S2 = (−5,−1, 1, 3, 4, 5, . . . )⇒ c(S2) = 0.This corresponds to the state e−5 ∧ e−1 ∧ e1 ∧ e3 ∧ e4 . . . with two negative energy statesoccupied (e−5 and e−1) and two “holes” in the positive spectrum (e0 and e2). Therefore,the net-charge of the state is zero.

The role of the basis (ej)j∈Z can be understood as a choice of a complete set of one-particlestates, used to characterize the particle content of the Fock space states (≈ Dirac seas). Ideally,this choice is determined, or at least motivated, by physical properties e.g. as the spectraldecomposition of a (or several commuting) self-adjoint operator(s) onH. Anyways, the sections(ΨS)S∈S will make a good basis for the Fock space.

Proposition 5.3.12 (Hermitian Form).Let V ⊆ Γ(DET ∗) be the complex vector space spanned by the sectionsξz ∈ Γ(DET ∗) | z = [(z, 1)] ∈ DET. Then

〈ξz, ξw〉 := ξz(w) := Φ(z, w) (5.3.8)

defines a Hermitian form on V, antilinear in the second component. This form is positivesemi-definite. The sections

ΨS

S∈S form a complete orthonormal set in V w.r.to 〈·, ·〉.

Proof. First we note that 〈ξz, ξw〉 = Φ(z, w) = det(z∗w) = det(w∗z) = 〈ξw, ξz〉.Then we recall that Φ(·, ·) is C-linear in the second entry and anti-linear in the first entry, butthe mapping w 7→ ξw is C-anti-linear. In conclusion, 〈·, ·〉 defines a sesquilinear form anti-linearin the second entry. It is easy to see that for S, S′ ∈ S

〈ΨS ,Ψ′S〉 = ΨS(w′S) = det(w∗Sw

′S) = δSS′

because w∗SwS = 1H≥−n , whereas w∗Sw′S has non-trivial kernel if S 6= S′.

Now, in finite dimensions, if A is a n×m matrix and B a m× n matrix with n ≤ m then

det(AB) =∑

(i)=(1≤i1≤...≤in≤m)

det(AP(i)) det(P(i)B)

where P(i) denotes the projection onto span(ei1 , . . . ein). As the Fredholm determinant ofan operator is the limes of the determinants of its restriction to n-dimensional subspaces as

72


n→∞, the above formula extends to the infinite-dimensional case yielding

〈ξz, ξw〉 = Φ(z, w) = det(z∗w) =∑

S∈S, c(S)=d

det((PHSz)∗) det(PHSw)

=∑S

det((w∗Sz)∗) det(w∗Sw) =

∑S

det((w∗Sz)) det(w∗Sw)

=∑S∈S

ΨS(z)ΨS(w) =∑S∈S〈ξz,ΨS〉〈ΨS , ξw〉

for ind(P+z) = ind(P+w) = d and 0 else. This shows that ΨS∈S is indeed a complete,orthonormal system in V. Finally, for general Ψ =

∑finite

αnξzn ∈ V we compute

〈Ψ,Ψ〉 =∑n,m

αn αm 〈ξzn , ξzm〉 =∑n,m

∑S∈S

αn αm 〈ξzn ,ΨS〉〈ΨS , ξzm〉

=∑S∈S

(∑n

αn〈ξzn ,ΨS〉) (∑

m

αm〈ΨS , ξzm〉)

=∑S∈S

∣∣∣∑n

αn〈ξzn ,ΨS〉∣∣∣2 ≥ 0

Thus, the Hermitian form is positive semi-definite. This finishes the proof.

Definition 5.3.13 (Fermionic Fock Space).Let V0 := v ∈ V | 〈v, v〉 = 0 be the null-space of (V, 〈·, ·〉).

We define the fermionic Fock space F = Fgeom to be the completion of V/V0 w.r.to 〈·, ·〉.Fgeom is an infinite-dimensional, complex, separable Hilbert space.

It can be written as the direct sum F =⊕c∈ZF (c) of Z “charge-sectors” built from holomorphic

sections with support in the connected component of DET ∗ over Grc(H).

The sections ΨSS∈S are a Fock basis of F . They define the Pflücker coordinates

F −→ `2,

ξz 7−→ (ξz(wS))S∈S = (〈ξz,ΨS〉)S∈S.(5.3.9)

In general, we write F for “the” fermionic Fock space and Fgeom, if we want to emphasize thatit is the Fock space from Def. 5.3.13, obtained from the geometric construction, as opposed todifferent constructions presented before.

Definition 5.3.14 (Pflücker Embedding).We have a natural embedding of Gr(H) into the projective Fock space P(Fgeom), given by

Gr(H) −→ P(Fgeom),

W 7−→ C · ξw = C · ξ([w, 1])(5.3.10)

where w is an admissible basis for W, called the Pflücker embedding.This gives precise meaning to our often employed intuition that polarizations correspond toprojective, decomposable states of infinitely many fermions.

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Proposition 5.3.15 (Action of GL0

res on DET0).GL

0

res has a natural action on DET defined by

µ0 : GL0

res ×DET0 −→ DET0,

([U,R], [W,w, λ]) 7−→ [UW,UwR−1, λ].(5.3.11)

Proof. We have to show that the action µ0 is well-defined.

i) DET0 is closed under the action of GL0

res:We know that W ∈ Gr(H)⇒ UW ∈ Gr(H) for U ∈ GLres. We still have to show:w admissible basis for W ⇒ UwR−1 admissible basis for UW .Obviously, im(UwR−1) = UW . Furthermore:

P+UwR−1 = P+UP+wR

−1 + P+UP−wR−1

= P+UP+P+wR−1 + P+UP−P−wR

−1

= P+UP+R−1︸︷︷︸

∈Id+I1(H+)

RP+wR−1︸︷︷︸

∈Id+I1(H+)

+P+UP−︸︷︷︸∈I2(H+)

P−wR−1︸︷︷︸

∈I2(H+)

∈ Id+ I1(H+)

since the product of two Hilbert-Schmidt operators is trace-class.Thus, UwR−1 is an admissible basis for UW .

ii) The definition is independent of the representative of [W,w, λ]:Let (W,w, λ) ∼ (W,w′, λ′) = (W,w L, λ/det(L)).We need (UW,UwR−1, λ) ∼ (UW,UwLR−1, λ/det(L)), for U and R as above.This is true because UwLR−1 = UwR−1(RLR−1) = UwR−1L′, withL′ = RLR−1 ∈ GL1(H+) and det(L′) = det(RLR−1) = det(L).

iii) The definition is independent of the representative of [(U,R)] ∈ GL0

res:Let [(U,R)] = [(U,R′)] ∈ GL

0

res. This means det(RR′−1) = 1. Setting L := RR′−1 ∈GL1(H+), we find UwR′−1 = UwR−1RR′−1 = UwR−1L, with det(L) = 1 and thus(UW,UwR−1, λ) ∼ (UW,UwR′−1, λ).

Construction 5.3.16 (Arbitrary Charges and the complete GLres).So far, we have defined the action of GL

0

res on DET0 and in fact, this is all we need. Extendingthe action to arbitrary “charges" is somehow tedious, yet pretty much straightforward. Again,we use the “shift”-operator σ and the structure of GLres as a semi-direct product Z n GL

0

res

with Z generated by the action of σ (see §4.1.2).

We define a Z−action on DET by

ϑ(n)([W,w, λ]) := [σn(W ), σnwσ−n, λ]. (5.3.12)

Note that ϑ(n) maps DETd into DETd−n for any d ∈ Z..

Now we extend the action µ0 defined above to an action µ : Z n GL0

res ×DET → DET by

µ(n, [A,R])∣∣DETd→DETd−n

:= ϑ(n− d) µ0(σd−n([A,R])) ϑ(d). (5.3.13)

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In particular, GL0

res acts on DETd by

µ[A,R] := ϑ(−d) µ0(σd[A,R]) ϑ(d). (5.3.14)

This is almost the same as the action of GL0

res on St(d)(H) descending to DET , only with Aacting from the left and (Rσd)−1 (instead of R−1) acting from the right.

Admittedly, this is not very pretty. Therefore, we will be so daring and drop the Z−indices forthe remainder of this chapter and use the simple notation for GL

0

res acting on the “zero-chargesector”, while still stating the results for the whole transformation group and the whole linebundle. If needed, the full expressions for arbitrary charges can be worked out in detail usingthe scheme we’ve just outlined.

Finally, we can tell how the central extension GLres(H) and Ures(H) defined in Chapter 4 acton the fermionic Fock space and everything fits together nicely.

Proposition 5.3.17 (Action of GLres on Fgeom).The action of GLres(H) on F is given by the adjoint action of GLres on DET as defined inProp. 5.3.15. This action restricts to a unitary representation of Ures(H) on the Fock space.Explicitely, the action of GLres(H) on F is defined by

µ∗ : GLres ×F −→ F ,

([U,R], ξz) 7−→ µ∗[U,R]ξz = ξz µ−1[U,R]

(5.3.15)

i.e. for w ∈ DET :

µ∗[U,R]ξz(w) = ξz(µ−1[U,R](w)) = ξz(U

−1wR)

= det(z∗U−1wR) = det(Rz∗U−1wRR−1)

= det((U−1∗zR∗)∗w) =: ξz′(w)

with z′ = (U∗)−1zR∗. In the unitary case, where [(U,R)] ∈ Ures(H) ⊂ GLres(H), i.e. U ∈Ures(H) and R ∈ U(H+), this simplifies to

ξz 7→ µ∗[U,R](ξz) = ξz′ , with z′ = UzR∗. (5.3.16)

Clearly, this defines a representation of Ures(H) which is indeed unitary, because⟨µ∗[U,R]ξz, µ

∗[U,R]ξw

⟩= det

((UzR∗)∗(UwR∗)

)= det(Rz∗wR−1) = det(z∗w) =

⟨ξz, ξw

⟩.

Generalization to arbitrary polarizations

The geometric construction can be immediately generalized to arbitrary polarization, the onlychallenge being that it complicates our notation.Let Pol(H) be the set of all polarizations of H, i.e. of all closed, infinite-dimensional subspaceswith infinite-dimensional orthogonal complements. In §2.1 we introduced the equivalence re-lation

V ≈W ⇐⇒ PV − PW ∈ I2(H),

defining a partition of Pol(H) into disjoint polarization classes C ∈ Pol(H)/≈.

75


Given a fixed polarization class C ∈ Pol(H)/≈, we choose a subspace W ∈ C correspondingto the splitting H = W ⊕W⊥. This polarization distinguishes I2(W ;W⊥) as a model-spacefor the manifold structure on C = [W ]≈, analogous to that of Gr(H) for H+. The resultingGrassmann manifold Gr0(H,W ) is a homogeneous space for

Ures(H; [W ], [W ]) = U ∈ U(H) | UW ≈W

= U ∈ U(H) | [PW − PW⊥ , U ] ∈ I2(H).

This group inherits a Lie group structure from its representation onW⊕W⊥, with the topologyinduced by the norm ‖U‖εW = ‖U‖+ ‖ [PW − PW⊥ , U ] ‖2.Furthermore, we define in analogy to Def. 5.3.4 an admissible basis for V ∈ Gr0(H,W ) asa bounded isomorphism w : W → V with PW w ∈ Id + I1(W ) (and generalize to arbitrarycharges). That is, the admissible bases of subspaces in [W ]≈ are those compatible with theidentity map on W . The set of all such admissible bases is the Stiefel manifold St(H,W ). Incomplete analogy to the construction above, we can now define the determinant bundle for Was the associated line bundle

DETW = DETW(St(H,W ),Gr(H,W ),det : GL1(H+)1(W )→ C

)and the Fock space FWgeom as the space of holomorphic sections in DET∗W, equipped with thescalar product 〈ξz, ξw〉 = det(z∗w).

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5.4. EQUIVALENCE OF FOCK SPACE CONSTRUCTIONS

5.4 Equivalence of Fock Space Constructions

The infinite wedge space construction of Deckert, Dürr, Merkl and Schottenloher is closelyrelated to the geometric construction of the Fock space as the space of holomorphic sectionin the dual of the determinant bundle. The infinite wedge spaces, however, remain ratherunimpressed by the underlying geometry and highlight the algebraic relations that seem moreessential to the physical problems.

The reader might have noticed that the admissible bases are for the geometric construction,what the seas are for the infinite wedge spaces. However, it is important to note one essentialdifference. For the construction of the infinite wedge space, we start with a polarization classand have the freedom to choose a suitable Dirac sea class. In the geometric construction, on theother hand, we have to fix a polarization within the polarization class (e.g. H+ ∈ Gr(H)); thispolarization then comes with a preferred choice for the class of admissible bases, induced by theidentity map on that particular subspace.10 By this means, the geometric Fock space genuinelycontains a distinguished state, whereas all states in the infinite wedge spaces come with equalrights. In other words, the geometric construction, as opposed to the infinite wedge spaceconstruction, naturally yields a Fock space with vacuum. (This is just the usual nomenclature,the preferred state may or may not actually play the role of a vacuum state in the physicaldescription). In particular, there are as many (different, but equivalent) geometric Fock spacesover a fixed polarization class, as there are polarizations, i.e. vacua. Two infinite wedge spacesF and F ′ over the same polarization class C differ by right-action with an operator R ∈ U(`),with F = F ′ ⇐⇒ R ∈ U1(`) (see Cor. 5.2.11).

So much for the abstract discussion. Let’s now formulate the relationship between Diracsea classes and admissible bases in a precise way.

Lemma 5.4.1 (Dirac Seas vs. Admissible Bases).

Any isometric Dirac sea Φ0 ∈ Seas⊥(H) with im(Φ0) = H≥−n gives rise to anisomorphism (

Seas⊥(H)/ ∼)3 S(Φ0)

∼=−−→ St(n)(H)

between its Dirac sea class and the n-th connected component of the Stiefel manifold.

More generally, let Φ0 ∈ Seas⊥(H) with im(Φ0) = W . Then Φ0 induces an isomorphismbetween its Dirac sea class and the Stiefel manifold

St0(H,W ) = w : W → H | PWw ∈ Id+ I1(W ) (5.4.1)

corresponding to the polarization W ∈ Pol(H).

Proof. Given a Φ0 : `→ H≥−n ⊂ H as above, consider the map

St(n)(H)→ S(Φ0); w 7→ w Φ0, (5.4.2)

This is a well-defined map into S(Φ0): If w is an admissible basis, w Φ0 is a map from ` toH and indeed w Φ0 ∼ Φ0 in Seas⊥(H), because

10Actually, in both cases, the choice determines only the charge-0 sector of the Fock space. Con-struction of the “full” Fock space requires a choice for every charge-sector, i.e. for every connectedcomponent of the polarization class.

77

5.4. EQUIVALENCE OF FOCK SPACE CONSTRUCTIONS

P+w = Φ0Φ∗0w has a determinant

⇒ Φ∗0 wΦ0 : `→ ` has a determinant

⇒ w Φ0 ∼ Φ0 in Seas⊥(H).

It remains to show that (5.4.2) is bijective: Let Φ ∼ Φ0 ∈ Seas(H). Definew := Φ Φ∗0

∣∣H≥−n

. w is an isomorphism H≥−n → H with im(w) = im(Φ) ∈ Gr(H)

and clearly w Φ0 = Φ. Finally, w is indeed an admissible basis i.e. in St(n)(H), becauseP+w = Φ0Φ∗0 ΦΦ∗0 has a determinant⇐⇒ Φ∗0Φ has a determinant⇐⇒ Φ ∼ Φ0.The general case works analogously.

Theorem 5.4.2 (Anti-Isomorphism of Fock spaces).Let Φ0 ∈ Seas⊥(H) with im(Φ0) = H≥−n. The infinite wedge space FS(Φ0) is anti-unitaryequivalent to F (n)

geom, the charge-n-sector of the fermionic Fock space constructed from Γ(DET ∗n).

Proof. Consider the map f : F (n)geom −→ FS(Φ0), defined by

ξw 7−→∧(w Φ0

), for w ∈ St(n) (5.4.3)

and anti-linear extension. This is well defined, since for L ∈ GL1(H+), we haveξzL = det(L) ξz by (5.3.7), and

∧(w L Φ0

)=∧(w Φ0Φ∗0L Φ0

)= det(Φ∗0L Φ0)

∧(w Φ0

)= det(L)

∧(w Φ0

).

It is also an (anti-) isometry, because⟨∧(z Φ0

),∧(w Φ0

)⟩FS(Φ0)

= det(Φ∗0z∗wΦ0) = det(z∗w) =

⟨ξz, ξw

⟩Fgeom

.

By construction of the infinite wedge space and by the previous Lemma, Dirac seas of the formwΦ0, w ∈ St(n)(H) span the entire Fock space FS(Φ0). We conclude that the two Fock spacesare anti-unitary equivalent.

Corollary 5.4.3 (Action of U0res on Infinite Wedge Spaces).

Let Φ0 and FS(Φ0) as above. We have a natural representation of U0res(H) on FS(Φ0) given by∧

U: U0

res(H) 3 [U,R] 7−→ LURR∗Φ0(5.4.4)

with RΦ0:= Φ∗0 RΦ0 ∈ U(`), i.e.

∧Φ

[U,R]−−−→∧(U Φ Φ∗0R

∗Φ0

).

Of course, this is the representation induced by the that on F via the anti-automorphism f:

F (n)geom

f

µ∗[U,R] // F (n)geom

f

FS(Φ0)

LURR∗Φ0// FS(Φ0)

(5.4.5)

Proof. It suffices to note that

f µ∗[U,R] (ξw) = f (ξ(UwR∗)) =∧(UwR∗ Φ0

)=∧(UwΦ0 Φ∗0 R

∗Φ0

)= LURR∗Φ0

f (ξw).

78

5.5. CAR REPRESENTATION ON FGEOM

5.5 CAR Representation on Fgeom

In this section, we discuss the relationship between the geometric Fock space, or equivalently,the infinite wedge spaces, and the Fock spaces studied in Section 5.1 as representation-spaces ofthe abstract CAR-algebra, generated by creation and annihilation operators. We will provide amore rigorous formulation of our initial argument that a description in terms of “particles” and“antiparticles” is equivalent to formulations involving a “Dirac sea” of infinitely many particles.

Unfortunately, at this point we’ll have to pay the price for bowing to different, contradictoryconventions, when we used H+ for the unperturbed Dirac sea, e.g. in §5.2, but also for thepositive energy electron states, e.g. in §5.1. We will circumvent this difficulty, simply by settingF+ := H− and F− := CH+ in the construction of the Fock representation, i.e. in:

F :=

∞⊕c

F (c); F (c) :=⊕

n−m=c

∧nF+ ⊗

∧mF−.

Thus, compared to Section 5.1, the roles of H+ and H− are interchanged. I hope the readerwill excuse this little blemish.

Construction 5.5.1 (Fock Space Isomorphism and Field Operator on Dirac Seas).

We fix a basis (ek)k∈Z of H such that (ek)k≤0 is an ONB of H− and an (ek)k≥0 ONBof H+. Let S be the set of sequences as in Def. 5.3.10. It consists of increasing sequencesS = (i0, i1, i2, . . .), containing only finitely many negative integers and all but finitely manypositive integers. Recall that by Prop. 5.3.12, the holomorphic sections ΨSs∈S defined inDef. 5.3.10, iii), form an orthonormal basis of the Fock space Fgeom.

We define an isomorphism between Fgeom and F = F(H+,H−) by

ΨS 7−→(ei0 ∧ . . . ∧ ein−1

)⊗(Cej0 ∧ . . . ∧ Cejm−1

)∈∧nF+ ⊗

∧mF− (5.5.1)

for S ∩ Z− = i0 < i1 < . . . < in−1 and N \ S = j0 < j1 < . . . < jm−1.

In particular, for S = (0, 1, 2, . . .),ΨS = Ψ0 is mapped to the vacuum state in F .

In other words, the states indexed by the negative integers in S are mapped to “electronstates” in F+ and the states indexed by the positive integers missing in S (the “holes”) arecharge-conjugated and mapped to “positron states” in F−.Obviously, the assignment is 1–to–1 and isometric. It also preserves charge-preserving, in thesense that states ΨS with c(S) = c, spanning the charge-c-sector of Fgeom, are mapped intoF (c), the charge-c-sector of F .

We can also introduce the equivalent of creation- and annihilation operators on Fgeom.For this, it is convenient to use the intuitive notation

ΨS = ei0 ∧ ei1 ∧ ei2 ∧ ei3 ∧ . . . .

for S = (i0, i1, i2, i3 . . .) ∈ S.

79

5.5. CAR REPRESENTATION ON FGEOM

Then, we can define the field operator (or rather its hermitian conjugate) Ψ∗ on Fgeom by

Ψ∗(ek) : ei0 ∧ ei1 ∧ ei2 ∧ ei3 ∧ . . . 7−→ ek ∧ ei0 ∧ ei1 ∧ ei2 ∧ ei3 ∧ . . . (5.5.2)

That is, Ψ∗(ek) maps ΨS with S = (i0, i1, i2, i3 . . .) to zero, if k ∈ Z is contained in S, otherwiseto (−1)j ΨS′ with S′ = (i0, i1, . . . ij−1, k, ij , . . .) ∈ S. By linear extension in the argument ofΨ∗(·), we get a linear map Ψ∗ : H → B(Fgeom).

It is easy to see that under the isomorphism defined above, this field operator acts just as theusual field operator (5.1.9), defined in terms of creation- and annihilation operators.The same is true for the formal adjoint Ψ : H → B(Fgeom), acting as

Ψ(ek) ei0 ∧ ei1 ∧ ei2 ∧ ei3 ∧ . . . =

(−1)j ei0 ∧ ei1 . . . ∧ eij−1

∧eij ∧ eij+1∧ . . . ; if k = ij

0 ; if k /∈ S

In particular, Ψ satisfies the canonical anti-commutation relations

Ψ(f),Ψ∗(g) = 〈f, g〉H · Id, ∀f, g ∈ H

and therefore defines a representation of the CAR-algebra on the Fock space Fgeom which isequivalent to the Fock representation on F .

80

Chapter 6

Time-Varying Fock spaces

We have seen that a unitary operator on H can be implemented on the fermionic Fock space ifand only if it satisfies the Shale-Stinespring condition [ε, U ] ∈ I2(H) i.e. if and only if it belongsto Ures(H). In the previous chapter, this result appeared in different disguises depending onthe construction of the Fock space, but there was no way around the fact itself. The dramaticconclusion seems to be that there is no well-defined time evolution in QED. By Ruijsenaar’stheorem 1.2.1, the Dirac-time evolution for a field A will not be in Ures(H), unless the spatialcomponent A of the vector potential vanishes. Physically, the reason is infinite particle cre-ation. Mathematically, this is reflected in the fact that a unitary transformation that doesn’tsatisfy the Shale-Stinespring condition, will leave the polarization class [H−] = Gr(H), overwhich the standard Fock space is constructed (here, we use again H = H+⊕H− as the spectraldecomposition and H− plays the role of the Dirac sea).

In the light of these results, Deckert et.al. concluded that the best we can do is to realizethe time evolution as unitary transformations between different Fock spaces over varying po-larization classes. The first analysis along those lines was probably done by Fierz and Scharf in[FiScha79], who studied time-dependent Fock space representations for QED in external fields.A similar construction, realized in more geometrical notions, is also known as the Fock spacebundle in the mathematical literature (see for example [CaMiMu00]).

Note that in this work we are not focusing on finding the weakest regularity condition onthe external vector potentials. For simplicity, we always assume smooth fields with compactsupport in space and time, but in fact most of the results can be generalized a a bigger classof interactions, where suitable fall-off properties for the fields are assumed. The reader mayconsult the cited publications for notes on those possible generalizations.

81

6.1. IDENTIFICATION OF POLARIZATION CLASSES

6.1 Identification of Polarization Classes

Given an external field in a form of 4-vector potential

A = (Aµ)µ=0,1,2,3 = (A0,−A) ∈ C∞c (R4,R4),

let UA = UA(t, t′), t, t′ ∈ R be the unitary Dirac time evolution determined by the Hamiltonian

HA = D0 + e

3∑µ=0

αµAµ. (6.1.1)

The question is the following: How can we determine polarization classes C(t) such that

UA(t1, t0) ∈ Ures

(H, C(t0);H, C(t1)

)(6.1.2)

for all t1, t0 ∈ R ? The answer is a generalization of Ruijsenaars theorem and was provided ina very neat, systematic way by Deckert, Dürr, Merkl and Schottenloher in [DeDuMeScho]. Wegive a brief summary of their results.

The unitary time evolution UA(t1, t0), t0, t1 ∈ R satisfies the equation

∂

∂tUA(t, t0) = −iHAUA(t, t0) = −i(D0 + V A(t)

)UA(t, t0) (6.1.3)

with the interaction-term

V A = e

3∑µ=0

αµAµ. (6.1.4)

In momentum representation, i.e. after taking the Fourier transform of the equation, D0 actsas a multiplication operator by the energy E(p) =

√|p|2 +m2 and the interaction potential

takes the form

V A(p, q) = e

3∑µ=0

αµAµ, (6.1.5)

where the Aµ, µ = 0, 1, 2, 3, now act as convolution operators, i.e.

(Aµψ)(p) =

∫R3

Aµ(p− q)ψ(q) dq, p ∈ R3, (6.1.6)

for ψ ∈ H and Aµ the Fourier transform of Aµ.

From (6.1.3) we can deduce:

∂

∂t

[U0(t1, t)U

A(t, t0)]

= −iU0(t1, t)[HA(t) −H0

]UA(t, t0) = −iU0(t1, t)V

A(t)UA(t, t0).

This is equivalent to the integral equation:

82


UA(t1, t0) = U0(t1, t0)− it1∫t0

U0(t1, t)VA(t)UA(t, t0)dt. (6.1.7)

We can read this as a fixed-point equation for UA(t1, t0), leading to the iteration

UA(t1, t0) = U0(t1, t0)− it1∫t0

U0(t1, t)VA(t)U0(t, t0) dt (6.1.8)

−t1∫t0

t∫t0

U0(t1, t)VA(t)U0(t1, t

′)V A(t′)U0(t′, t0) dt′dt + ... (6.1.9)

known as the Born series. We need to control the odd part UAodd = UA

+− + UA−+ of the

time evolution that is responsible for “pair creation” and for the time evolution leaving thepolarization class. The free Dirac time evolution U0(t1, t0) = e−iD0(t1−t0) is diagonal w.r.tothe polarizationH = H+⊕H−, so that the relevant contributions in the perturbation expansionarise from ∫

U0[V A+− + V A

−+]U0. (6.1.10)

Now, for a time-independent electromagnetic potential A ∈ C∞c (R3,R4), Deckert et. al. definea bounded operator QA with integral kernel (in momentum-space)

QA(p, q) :=V A−+(p, q)− V A+−(p, q)

(E(p) + E(q)). (6.1.11)

It can be readily computed that (6.1.10) can be written as

i

t1∫t0

U0(t1, t)[V A(t)+− + V A(t)−+

]U0(t, t0)dt

= QA(t1)U0(t1, t0)− U0(t1, t0)QA(t0)−t1∫t0

U0(t1, t)QA(t)U0(t, t0)dt

(6.1.12)

with QA(t) = QA(t) ([DeDuMeScho], Lemma III.6). Now the idea is to use the Q-operators to“renormalize” the time evolution in such a way that the Hilbert-Schmidt norm of the odd-partremains finite. Equivalently (in a sense that will be discussed later), we can use these operatorsto identify the polarization classes into which the (unaltered) Dirac time evolution is mapping.Indeed, since QA(t) is skew-adjoint, eQ

A(t)

is unitary and it can be shown that

e−QA(t1) UA(t1, t0) eQ

A(t0) ∈ Ures(H) (6.1.13)

for all A ∈ C∞c (R4,R4) and all t0, t1 ∈ R, i.e. the odd parts are Hilbert-Schmidt operators.More generally, Deckert et. al. proof the following crucial theorem:

83


Theorem 6.1.1 (Identification of Polarization Classes).Let A ∈ C∞c (R4,R4). The operators eQ

A(t)

have the following properties:

i) Setting C[A(t)] := [eQA(t)H−] it is true that

UA(t1, t0) ∈ U0res

(H;C[A(t0)], C[A(t1)]

)(6.1.14)

for all t0, t1 ∈ R.

ii) For two static potentials A = (A0,−A) and A′ = (A′0,−A′) ∈ C∞c (R3,R4) we find

[eQA

H−]≈0= [eQ

A′

H−]≈0⇐⇒ A = A′. (6.1.15)

It follows that the polarization class is completely determined by the spatial component of thevector potential A at any fixed time.

In more physical terms, the theorem says that the polarization classes C[A(t)]

• depend only on the magnetic part of the interaction.

• depend on A instantaneously in time and not on the history of the system.

This justifies the notation C[A(t)] for the respective polarization classes, or even C[A] for astatic field A. (Note that in our convention, C[A] = C[A = (0,−A)].)We observe that Ruijsenaars Theorem 1.2.1 follows immediately as a special case of (6.1.15).Furthermore, the following important result now follows as a simple corollary:

Theorem 6.1.2 (Implementability of the S-matrix).Let A ∈ C∞c (R4,R4) and UA

I the corresponding unitary time evolution in the interaction picture(cf. §8.1.2). Then, the scattering matrix

S := limt→∞

eitD0UA(t,−t)eitD0 = limt→∞

UAI (t,−t) (6.1.16)

is in Ures(H) and can be implemented as a unitary operator on the (standard) Fock space.

Proof. Since the interaction potential has compact support and thus, in particular, compactsupport in time, we have A(t) = 0 and hence eQ

A(t)

= 1 for all |t| large enough. In particular,[eQ

A(t)H−]≈0= [H−]≈0

for all |t| large enough. By the previous theorem it follows that

UAI (T,−T ) = U0(0, T )UA(T,−T )U0(−T, 0) ∈ U0

res(H; [H−], [H−]) = U0res(H)

for all T large enough and hence S ∈ U0res(H).

84

6.2. SECOND QUANTIZATION ON TIME-VARYING FOCK SPACES

6.2 Second Quantization on Time-varying Fock Spaces

We present the recipe for second quantization of the time evolution on time-varying Fock spacesaccording to Deckert et. al., as described in [DeDuMeScho]. The main ingredient is Theorem5.2.9, the abstract version of Shale-Stinespring.Let A = (Aµ)µ=0,1,2,3 = (A0,−A) ∈ C∞c (R4,R4) be an external field and UA(t, t′) the corre-sponding Dirac time evolution.

• Let C(t) = C[A(t)] ∈ Pol(H)/ ≈0 be the polarization classes identified above. By 5.2.9,C(t) is uniquely determined by the spatial component of the vector potential at thattime and

UA(t, t′) ∈ U0res

(H;C(t′), C(t)

), ∀t, t′ ∈ R.

• For every t ∈ R, choose a Dirac sea class S(t) ∈ Ocean(C(t))/ ∼.For the geometric construction, we choose polarizations W (t) ∈ C(t), instead.We should demand that S(t), respectively W (t) depends only on A(t) or even just onA(t). In particular, it is reasonable to demand that we stay in the initial (“standard”)Fock space, whenever the interaction is turned off.

• We construct a family of Fock spaces (FS(t))t as infinite wedge spaces over the Diracsea classes S(t). Equivalently, we can use the geometric construction where admissiblebases are those compatible with the identity map on W (t).

• By Theorem 5.2.9, we can implement UA(t1, t0) as a unitary map between the Fockspaces FS(t0) and FS(t1). There exists R ∈ U(`) with UA(t1, t0)S(t0)R = S(t1) andtherefore

LUA(t1,t0)RR : FS(t0) → FS(t1) (6.2.1)

is a unitary map between the Fock spaces FS(t0) and FS(t1).

• By Lemma 5.2.8, two right operations implementing UA(t1, t0) differ by an operator inU(`) ∩ Id + I1(`). The induced transformation between FS(t0) and FS(t1) can differ bythe determinant of such an operator, i.e. by a complex phase ∈ U(1).However, transition probabilities are well-defined : if we have an “in-state” Ψin ∈ FS(t0)

and an “out-state” Ψout ∈ FS(t1), the transition probability

|〈Ψout,LUA(t1,t0)RR Ψin〉|2 (6.2.2)

is independent of the choice of R ∈ U(`).

This is the scheme for implementation of the unitary time evolution on time-varying Fockspaces in its most general form. The formal clarity of the results (that we might or might nothave been able to convey), could seduce us into too much optimism. In fact, the construction,as it stands so far, is neither practicable nor physically meaningful. In practice, we wouldrather make a “global” choice of Fock spaces, preferably one that depends only on the externalpotential, locally in time. For example, starting with any sea Φ ∈ Ocean(H−) we can set

S(t) := [eQA(t)Φ] ∈ Ocean

(C[A(t)]

)/ ∼ (6.2.3)

85


which depends only on A(t) at time t. By Thm. 6.1.1,(FS(A(t))

)tdefines a suitable family of

Fock spaces for any external field A. Note, however, that this choice is merely a convenientone and not motivated by physical insight.

Also note that the implementations we obtain will not readily yield a time evolution. Thatis, arbitrary choices of the right-operations by R = R(t1, t0) will in general not be “compatible”with each other, so that for t2 ≥ t1 ≥ t0 we are going to find that

LUA(t2,t1)RR(t2,t1)LUA(t1,t0)RR(t1,t0) 6= LUA(t2,t0)RR(t2,t0).

This problem is discussed in Section 8.2 under the more general theme of causality. There, itis proven that compatible choices, preserving the semi-group structure of the time evolution,are indeed possible.

6.2.1 Particle Interpretation

The most important question, however, concerns the physical content of the presented solution.It is yet unclear what physical quantities are meaningful in the context of time-varying Fockspaces and how physical states represented by vectors in different Fock spaces can be compared.Considering our discussion up to this point, it seems natural to ask about the particle contentof a given state:

“How many particles and anti-particles are present at time t ?”.

It is clear, though, that the sea classes S(t) and the resulting infinite wedge spaces alone donot provide enough structure to define how the Dirac sea is filled i.e. how many electrons andpositrons a given state contains. Thus, to get a well-defined answer, we have to distinguishinstantaneous “vacua” to compare our states to. Recall that the construction of the geomet-ric Fock space already contains, or rather requires such a distinguished state by choice of apolarization within the correct polarization class, defining the Stiefel manifold of admissiblebases. In the infinite wedge spaces, there is no such preferred state and a vacuum alwaysconstitutes an additional choice. It is however convenient to fix the various Fock spaces bychoosing suitable representatives of the Dirac sea classes, i.e. seas with images in the rightpolarization class. These “reference states” then may or may not represent a physical vacuum.

It is usually convenient to specify the (instantaneous) vacuum states only projectively,i.e. by identifying the Dirac sea as a polarization W (t) ∈ C[A(t)], ∀t ∈ R. Such a po-larization W ∈ Pol(H) specifies a unique geometric Fock space FWgeom (see Section 5.3).It also defines a unique GNS-representation of the CAR-algebra with a GNS-vacuum (seeSection 5.1.3) or, more down to earth, a unique Fock-representation via the field operatorΨW (f) = a(PW f) + b∗(PW⊥f) (see Prop. 5.1.14). For the infinite wedge space construction,it is not sufficient to specify the vacuum state only projectively. A fixed polarization within apolarization class does not specify an infinite wedge state or even a Fock space in a canonicalway. Two seas Φ and Ψ with im(Φ) = im(Ψ) = W can differ by a transformation R ∈ U(`)

(acting from the right) with Φ ∼ Ψ = ΦR if and only if R ∈ U1(`) (cf. Lemma 5.2.8). Theresulting Fock spaces are however isomorphic by RR : FS(Φ) → FS(Ψ).

86


Without any additional insight into the fully-interacting theory and the microscopic dy-namics of the Dirac sea, the only suggested choice for instantaneous vacua is given by spectraldecomposition with respect to the Hamilton including the external interaction potential. Thatis, given an external field A ∈ C∞c (R4,R4), we denote by P

A(t)+ and P

A(t)− the orthogonal

projections onto the positive and negative spectral subspaces of the (static) Hamiltonian

HA(t) = D0 + e

3∑µ=0

αµAµ(t)∣∣∣t=const.

.

We assume that 0 is not in the spectrum of HA(t) or at most an isolated eigenvalue of finite-multiplicity and include the corresponding eigenspace into PA(t)

− (H), let’s say, to get an unam-biguous prescription. Then, the following is true:

Theorem 6.2.1 (Fierz, Scharf 1979).Under the conditions formulated above, it is true that

PA(t)− (H) ∈ C[A(t)], ∀t ∈ R

or, equivalently,P

A(t)− − PUA(t,−∞)H− ∈ I2(H),

where by t′ = −∞ we mean a large negative t′ outside the support of t→ A(t).

Since we know from Thm. 6.1.1 [Identification of Polarization classes] that the polarizationclass depends on the spatial part of the field only, we may also take the spectral decompositionfor the Hamiltonian with A0 set to zero. Then, we have (using β2 = 1 and αk, β = 0):

(HA)2 =[−iα · (∇− ieA) + βm

]2=[−iα · (∇− ieA)

]2+m2 1

≥m2 1

so that Spec(HA) ⊆ (−∞,−m] ∪ [+m,+∞), i.e. we have a nice mass-gap between the Diracsea and the excited electron states.

The problem with both alternatives is that the vacuum is obviously not Lorentz-invariant.A Lorentz transformation will change the external field and therefore the energy spectrum ofthe Hamiltonian. This would mean that a vacuum state, defined with respect to a distinctreference frame, would look like a multi-particle state for an observer in a different referenceframe. It is on this basis that Scharf and Fierz finally reject the particle interpretation fora time-dependent interaction and conclude that “the notion of particles has only asymptoticmeaning" ([FiScha79], p. 453). We will discuss this conclusion in the final chapter.

6.2.2 Gauge Transformations

Given a unitary time evolution UA(t, t′), the polarization classes C(t) = C[A(t)] are deter-mined by the spatial components of the A-field, defining the interaction potential. But thisquantity is blatantly not gauge-invariant. Therefore, neither the Shale-Stinespring conditionUA(t, t′) ∈ Ures(H), nor the procedure of second quantization on time-varying Fock spaces isgauge-invariant. This might seem rather suspicious at first, since QED, if anything, is famous

87


for being a gauge-theory. But the right conclusion to draw is that it’s unclear, how the pre-sumed gauge invariance of the second quantized theory has to be understood. It turns outthat the problem with gauge transformations is the same as with the time evolution. Smoothgauge transformations

G 3 g : Ψ(x)→ eiΛg(x)Ψ(x), Λg ∈ C∞c (R3,R) (6.2.4)

do not satisfy the Shale-Stinespring condition [ε, g] ∈ I2(H) unless they are constant, and aretherefore not implementable as automorphisms on a fixed Fock space [MiRa88].

More precisely, the following is true:

Theorem 6.2.2 (Gauge Transformations, [DeDuMeScho]Thm. III.11).

Let G denote the space of smooth gauge transformations as in (6.2.4)

By [Thm. 6.1.1, ii)] it is justified to write [eQAH−] =: C[A] etc. for the polarization classes,

as they are completely determined by the spatial part of the A-field.Then, for all g ∈ G it is true that

g = eiΛg ∈ U0res

(H;C[A], C[A +∇Λg]

)(6.2.5)

for any A ∈ C∞c (R3,R3).

Of course, this is what must be true in order to be consistent with our previous results.

Proof. Let g ∈ G a smooth gauge transformation, corresponding to the multiplication operatoreiΛg on L2(R3,C4). Let f : R → [0, 1] be a smooth cut-off function with f(0) = 0, f(1) = 1

and compact support in, let’s say, [0, 2]. To apply the machinery developed so far, we make gtime-dependent by setting g(t) := ei f(t)Λg . Let A(t, x) = (0,−A(x)) a static vector-potential.Under the gauge transformation g, the vector potential transforms as Aµ → Aµ − ∂µ(f(t)Λg),so that A becomes

A(t, x) =(−f ′(t)Λg(x),−A(x)− f(t)∇Λg(x)

).

The time evolution transforms as

g(t1)UA(t1, t0)g∗(t0) = U A(t1, t0), t0 ≤ t1 ∈ [0, 1].

Note that the gauge transformation is fully “turned on” at t = 1 and “turned off” at t = 0. ByThm. 6.1.1 [Identification of Polarization classes], we have

UA(1, 0) ∈ U0res

(H;C[A], C[A]

)U A(1, 0) ∈ U0

res

(H;C[A], C[A +∇Λg]

)and therefore, by (2.1.4):

eiΛg = g(1) = U A(1, 0)UA(0, 1) ∈ U0res

(H;C[A], C[A +∇Λg)

].

88


In the spirit of this chapter, we can therefore realize gauge transformation not as unitary op-erations on a Fock space, but as a transformation of the Fock spaces themselves. In this sense,the theory might become “gauge-covariant” rather than “gauge-invariant” in the naive sense.We will come back to the problem of gauge invariance in Section 8.4.

Remark 6.2.3 (Renormalized Determinants).Mathematically, the gauge transformation group G and its representations have been studiedvery thoroughly. We know that smooth gauge transformations g ∈ G cannot be in GLres(H),i.e. their odd-part w.r.to the splitting H = H−⊕H− is not in the Hilbert-Schmidt class (unlessthey are constant). However, one can show that they are contained in a higher Schatten class;actually, the Schatten index depends on the dimension of the space-time manifold. In [MiRa88]it is proven that on d+ 1-dimensional space-time, for odd d,

[ε, g] ∈ I2p(H) ∀ g ∈ G ⇒ p ≥ (d+ 1)/2. (6.2.6)

Hence, G embeds in a smaller group

GLp(H) := A ∈ GL(H) | [ε, A] ∈ I2p(H) ⊂ GL(H).

GL1(H) = GLres(H) is good enough in 1+1-dimensions. For higher dimensions, in particular3+1 dimensional QED, it is possible in a certain sense to generalize our construction to arbi-trary Schatten index p. In particular, it is possible to extend the definition of the Fredholmdeterminant to operators in Id+ Ip(H) by including a suitable “renormalization” term. Then,one can construct central extensions GLp(H) of GLp(H) that reduce to GLres(H)→ GLres forp = 1 and act on smooth1 sections in the dual of a generalized determinant bundle DETp. See[Mi] for details on these constructions. But apart from being rather tedious, this solution isunsatisfying for a different reason: The action of G ⊂ GLp(H) is not unitary. Actually, it isknown that GLp(H) does not have any faithful unitary representations for p > 1 (see [Pick89]).

1The action is not any more holomorphic for p > 1, see [Mi].

89

Chapter 7

The Parallel Transport ofLangmann & Mickelsson

We have seen that the central extensions GLres(H)→ GLres(H) and Ures(H)→ Ures(H) carrythe structure of principle fibre bundles. This allows us to equip them with a bundle-connectionas an additional geometric structure. Our aim is to use parallel transport with respect to thisconnection to lift the unitary time evolution to the Fock space, or at least to fix the phase ofthe second quantized S-matrix in a well-defined manner. We will follow [LaMi96] and define asuitable connection not on Ures(H) but on its complexification GLres(H). However, the con-struction restricts easily to the unitary case.

Recall that a connection Γ on a principle bundle is a distribution in its tangent bundle, dis-tinguishing tangent vectors that are called horizontal (to the base-manifold). This distributionis complementary to the space of vertical vector fields, defined as the kernel of Dπ, where πis the projection onto the base-manifold and Dπ its differential map. That is, vertical vectorsare vectors along the fibres of the bundle. A connection is a geometric structure that allowsus to lift paths from the base-manifold (here: Ures or GLres) to the bundle in a unique way, bydemanding that the tangent vectors to the lifted path are always horizontal. These horizontallifts define parallel transport in the principle bundle.1 Furthermore, recall that connections ona principle bundle are in one-to-one correspondence with connection one-forms with values inthe Lie algebra of the structure group. For every connection Γ there exists one – and only one– connection form with Γ = ker(Θ). A nice treatment of connections and parallel transport onprinciple bundles can be found, for instance, in [KoNo].

Throughout this chapter, a little care is required with the fact that we’re working oninfinite-dimensional manifolds. But Banach manifolds, as the ones we’re dealing with, aregenerally pretty well-behaved. One essential point is that on Banach manifolds (as opposed tomanifolds modeled on infinite-dimensional Fréchet spaces, for example) the Inverse MappingTheorem holds and thus the local calculus works almost exactly as in the finite-dimensionalcase.Note, however, that we avoid all expansions in local coordinate frames, as are often used forthe analogous proofs in the finite-dimensional case. Fundamentals of Differential Geometry on

1If π : P →M is a principle bundle overM with a connection, the parallel transport of p ∈ P alonga curve γ in M starting in π(p) is the end-point of the horizontal lift of γ with starting point p ∈ P .

91

7.1. THE LANGMANN-MICKELSSON CONNECTION

Banach manifolds are presented in [Lang].

7.1 The Langmann-Mickelsson Connection

7.1.1 The Maurer-Cartan Forms

Let G be a Lie group. The group operation on G defines smooth actions of G on itself bymultiplication from the left or from the right. For g ∈ G we denote by Lg the action from theleft, i.e.

Lg : G→ G, Lg(h) := g · h (7.1.1)

and by Rg the action from the right

Rg : G→ G, Rg(h) := h · g. (7.1.2)

The differentials of these maps (the push forwards) as maps between tangent-spaces are thenisomorphisms:

(Lg)∗ : ThG→ TghG

(Rg)∗ : ThG→ ThgG.(7.1.3)

The corresponding pull-back maps on differential forms will be denoted by (Lg)∗ and (Rg)

∗.Recall that a vector field X ∈ X(G) is called left-invariant if (Lg)∗Xh = Xgh, ∀g, h ∈ G andright-invariant if (Rg)∗Xh = Xhg, ∀g, h ∈ G.

G can also act on itself by conjugation:

cg : G→ G, cg(h) := LgRg−1h = ghg−1. (7.1.4)

The mapAd : G→ GL(Lie(G)), g 7→ cg (7.1.5)

yields a natural representation of G on it’s own Lie algebra called the adjoint representation.The kernel of Ad equals the center of G.2

The Lie algebra g := Lie(G) of a Lie group G is usually defined as the space of left-invariantvector fields with the Lie bracket given by the commutator of vector fields. This space iscanonically isomorphic to the tangent space above the identity, i.e. g ∼= TeG, for e, the neutralelement in G. In the following, we make use of this identification.

Definition 7.1.1 (The Maurer-Cartan Forms).The left Maurer-Cartan form, also called the canonical left-invariant one-form, is a one-formon G with values in the Lie algebra. It’s defined by:

ωL(g) : TgG −→ TeG ∼= g ;

V 7−→ (Lg−1)∗ V(7.1.6)

2In a Matrix representation, G → GL(V ),Lie(G) → End(V ) and Adg(X) = gXg−1. Thus, one canthink of the adjoint representation in the following way: If the elements of a vector space V (or a vectorbundle with fibre V) transform under a symmetry g ∈ G via an action of G on V, the endomorphismsof V transform by the corresponding adjoint action.

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Analogously, we define the right Maurer-Cartan form by:

ωR(g) : TgG −→ TeG ∼= g ;

V 7−→ (Rg−1)∗ V(7.1.7)

Obviously, the Maurer-Cartan forms are constant on loft-invariant, respectively, on right-invariant vector fields. Furthermore, ωL(e) = ωR(e) = IdTeG.

In a matrix representation, they can be written as

ωL = g−1dg and ωR = dg g−1, (7.1.8)

respectively. We state a few less obvious properties of the Maurer-Cartan forms.

Lemma 7.1.2 (Properties of the left Maurer-Cartan form).The left Maurer-Cartan form ωL has the following properties:

i) ωL is left-invariant, i.e. (Lg)∗ωL = ωL

ii) (Rg)∗ωL = Adg−1 ωL, ∀g ∈ G

iii) dωL + 12 [ωL ∧ ωL] = 0

The right Maurer-Cartan form ωR has the following properties:

I) ωR is right-invariant, i.e. (Rg)∗ωR = ωR

II) (Lg)∗ωR = Adg ωR, ∀g ∈ G

III) dωR − 12 [ωR ∧ ωR] = 0

Proof. Let ω denote the left Maurer-Cartan form.

i) For V ∈ ThG and g ∈ G we have

(L∗g ωgh)(V ) = ωgh((Lg)∗V ) = L(gh)−1∗Lg∗V = Lh−1∗Lg−1∗Lg∗V = Lh−1∗(V ) = ωh(V )

ii) (R∗g ωhg)(V ) = ωh(Rg∗V ) = L(hg)−1∗Rg∗V = Lg−1∗Lh∗Rg∗V = Adg−1 ωh(V )

iii) dω(X,Y ) = X(ω(Y ))− Y (ω(X))− ω[X,Y ], for X,Y ∈ X(G).So, on left-invariant vector fields the identity is true, becausedω(X,Y ) = −ω[X,Y ] = −[ω(X), ω(Y )] = − 1

2 [ω(X)∧ω(Y )]. But the expression is bilinearin smooth functions (or in other words: tensorial) and thus depends on the vectors onlypointwise. Therefore, the identity holds for all vector fields.

The proofs for the right-invariant form work analogously. The different sign in III) as comparedto iii) comes from the fact that the commutator on right-invariant vector fields equals minusthe commutator on the corresponding left-invariant vector fields which – by convention – definesthe Lie bracket.

As the Lie algebra of a Lie group G is, by convention, usually defined as the vector space ofleft-invariant vector fields on G, the left Maurer-Cartan form is the more canonical object.However, it will turn out that the right-invariant form is better suited to our purposes.

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7.1.2 Connection Forms

Now, we have all ingredients for defining a connection on the principle C×-bundleGLres

π−−→ GLres. Recall that the central extension of Lie groups (4.1.2)

1 −→ C× ı−→ GLres(H)π−−→ GLres(H) −→ 1

induces a central extension of the corresponding Lie algebras

0 −→ C ı−→ g1π−→ g1 −→ 0. (7.1.9)

Furthermore, every (local) trivialization of GLres about the identity defines an isomorphism ofvector spaces g1

∼= g1 ⊕ C. In particular, we have the local trivialization φ defined in (4.1.6)as, arguably, the most natural choice and we use it to identify

φ : g1

∼=−−→ g1 ⊕ C.

With all this in our hands, a quite natural connection on GLresπ−−→ GLres can be readily

defined. The (left) Maurer-Cartan form ωL on GLres is a one-form on GLres with values inthe Lie algebra g1 = Lie(GLres). A connection, however, is defined by a one-form with valuesin the Lie algebra of the structure group which in our case is Lie(C×) = C. Thus, we simplyset ΘLM := prC ωL, where prC is the projection onto the C-component with respect to thesplitting g ∼= g⊕ C. This is indeed a connection one-form.

Definition 7.1.3 (Langman-Mickelsson Connection).Let ωL be the left Maurer-Cartan form on the Lie group GLres. The one-form

ΘLM := prC ωL ∈ Ω1( GLres;C) (7.1.10)

is a connection one-form on π : GLres(H)→ GLres(H).

We will call it the Langmann-Mickelsson connection.

In [LaMi96], Langmann and Mickelsson use this connection for a second quantization ofthe S-matrix by parallel transport in the GLres-bundle. We propose a slightly modified version,using the right-invariant Maurer-Cartan form, instead of the left-invariant form. The advantagewill become clear when we carry out the lift of the time evolution. We prove that the one-formsdefined in this way are indeed connection forms. The proof for the left-invariant form is almostidentical.

Proposition 7.1.4 (Connection on GLres).Let ωR be the right Maurer-Cartan form on the Lie group GLres. Then,

Θ := prC ωR ∈ Ω1( GLres;C) (7.1.11)

defines a connection one-form for the principle C× bundle π : GLres(H)→ GLres(H).

Proof. For an element x in the Lie algebra C of C×, we denote by x∗ the fundamental vectorfield, generated by x via its exponential action on the Lie group, i.e. by the flowΘt(p) = p · exp(tx). We need to show (cf. [KoNo], Prop. 1.1):

94


i) Θ(x∗) = x for all x in Lie(C×) ∼= C.

ii) R∗λΘ = Adλ−1Θ = Θ for all λ in the structure group C×.

In our case, the structure group happens to be abelian, so that Adλ−1 ≡ Id.ii) is trivial, because ωR is invariant even under the right-action of the entire GLres on itself,in particular under the right-action of ı(C×) ⊂ GLres. Thus:

R∗λ Θ = prC R∗λ ωR = prC ωR = Θ = Adλ−1Θ, ∀λ ∈ C×.

For the first property, let x ∈ C. Then:

x∗(p) =d

dt

∣∣∣t=0

p · ı(exp(tx)) = (Lp)∗ı∗(x)

⇒ ωR(x∗)|p = (Rp−1)∗(Lp)∗ ı∗ x = ı∗ x

⇒ Θ(x∗)|p = prC ı∗ x = x, ∀ p ∈ GLres(H).

where we have used that ı maps into the center of GLres(H) and also thatΘ ı(c) = (1, c) ∈ GLres × C×, which implies prC (ı∗)|e = Idg1

.

Thus, Θ is a connection one-form and defines the connection

ΓΘ := ker Θ ⊂ TGLres (7.1.12)

on the principle bundle GLres(H).

We can define an analogous connection on Ures(H) → Ures(H). However, we follow[LaMi96] in discussing the connection on the complexification GLres(H) → GLres(H). Apartfrom being more general, this has the advantage that the the local section in GLres(H) is“nicer” which simplifies computations in local coordinates. Anyways, our construction restrictsimmediately to the unitary case.

Proposition 7.1.5 (Curvature of the Connection).The curvature Ω of the connection Θ corresponds to the Schwinger cocycle (4.1.10) in thefollowing sense: If X, Y are right-invariant vector fields on GLres, identified with elements ofthe Lie algebra g1 of GLres then

Ω(X, Y ) = c(X,Y ) (7.1.13)

for X := Dπ(X), Y := Dπ(Y ) ∈ g1.3 Since Ω is a two-form and X(GLres) is spanned (asa C∞-module) by the right-invariant vector fields, the curvature of the bundle is completelydetermined by this identity.

Proof. Let X, Y be left-invariant vectorfields corresponding to (X,λ1), (Y, λ2) ∈ g1 ⊕ C ∼= g1,respectively. The curvature 2-form can be computed as Ω = dΘ + 1

2 [Θ ∧ Θ]. Since Θ takesvalues in the abelian Lie algebra C, the second summand is zero and Ω equals dΘ.

3The analogous computation for ΘLM would yield minus the Schwinger cocycle. In [LaMi96] theremight be a sign error or a different convention for the cocycle might be used.

95


We recall from Section 3.5 and Prop. 4.1.8 that under the isomorphism g1∼= g1 ⊕ C induced

by the local trivialization φ, the Lie bracket on g1 is the Lie bracket on g1 plus the Schwinger-cocycle. Thus, we get:

Ω(X, Y ) = dΘ(X, Y ) = prC(dω(X, Y )) = prC([ω(X), ω(Y )])

= prC([X, Y ]) = prC(([X,Y ], c(X,Y )

)= c(X,Y ).

7.1.3 Local Formula

Let’s compute an explicit formula for the connection w.r.to the coordinates defined by thelocal section (4.1.5). Let γ : t 7→ [(g(t), q(t))], t ∈ [−ε, ε] be a C1-curve in GLres(H). We write[(g, q)] for γ(0) = [(g(0), q(0)] and φ for the local trivialization (4.1.6) on π−1(U). Note thatthe connection form Θ can be expressed as:

Θg = prC(dg g−1) = prC DR−1g IdTgGLres

.

Thus, we compute:

Θ(γ(0)) =d

dt

∣∣∣t=0

prC φ([g(t)g−1, q(t)q−1]

)=d

dt

∣∣∣t=0

det[(g(t)g−1)−1

++(q(t)q−1)]

=− tr[(g(0)g−1)++ − q(0)q−1

].

Writing

g(t) =

(a(t) b(t)

c(t) d(t)

)and g−1(t) =

(α(t) β(t)

γ(t) δ(t)

)

w.r.t. the splitting H = H+ ⊕H−, the formula becomes

Θ = − tr[daα+ db γ − dq q−1

]. (7.1.14)

The corresponding expression for the left-invariant connection can be computed analogouslyand equals

ΘLM = − tr[α da+ β dc − q−1 dq

]. (7.1.15)

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7.2. PARALLEL TRANSPORT IN THE GLRES(H)-BUNDLE

7.2 Parallel Transport in the GLres(H)-bundle

We are actually interested in parallel transport in the GLres−bundle. The bundle-econnectiondefines a “horizontal” distribution in the tangent bundle of GLres(H) which gives us the notionof horizontal lifts of paths from GLres(H) to GLres(H). A horizontal lift is a path in GLres(H)

that projects to the original path in GLres(H) and whose tangent vector is always “horizontal”to the base manifold. So, if we think of the unitary time evolution as a differentiable pathin Ures(H) ⊂ GLres(H), the horizontal lift of that path will correspond to a continuous (evendifferentiable) choice of implementations on the fermionic Fock space.

We can use the local expression (7.1.14) for the connection form to compute an explicitformula for parallel transport inside the domain W of the local section (4.1.5):Let g(t) be a path in U ⊂ GLres(H), −T ≤ t ≤ T , with g(−T ) = 1. The lift g(t) = [(g(t), q(t))]

in GLres(H) is horizontal if and only if

tr[a(t)α(t) + b(t) γ(t)− q(t) q−1(t)

]≡ 0.

Formally, this impliestr(q(t) q−1(t)

)= tr

(a(t)α(t) + b(t) γ(t)

).

Identifying the LHS as the logarithmic derivative of det(q(t)) we can write

det(q(T )) = exp[ T∫−T

tr(a(t)α(t) + b(t) γ(t)) dt].

Also, formally: det(a(T )) = exp[ T∫−T

tr(a(t) a−1(t)) dt].

Individually, the traces do not converge but put together the trace converges and gives:

det[a−1(T )q(T )

]= exp

[ T∫−T

tr[a(t) (α(t)− a−1(t)) + b(t) γ(t)

]dt]. (7.2.1)

In the local trivialization φ onW ⊂ GL0res(H), this is precisely the C-component of the horizon-

tal lift g(T ) in (4.1.6). In other words, parallel transport in the GLres(H)-bundle correspondsto multiplication by the right-hand-side of (7.2.1) in local coordinates. Note that the expres-sion (7.1.14) is valid everywhere, while (7.2.1) makes sense only in the neighborhood W of theidentity, where a is invertible.The corresponding expression for the Mickelsson-Langmann connection is

det[a−1(T ) q(T )

]= exp

[ T∫−T

tr[(α(t)− a−1(t))a(t) + β(t)c(t)

]dt]. (7.2.2)

For a unitary path, these factor corresponds to the phase of the lift of g(T ) up to normalization(cf. Lemma 4.1.9). We have said that the structure defined on GLres restricts directly to theunitary case. Still, in case someone suspects hand-waving here, we show by explicit computa-tion that the lift of a unitary path doesn’t leave Ures(H) = GLres(H) ∩

(U(H)×U(H+)

).

97

7.3. CLASSIFICATION OF CONNECTIONS

Lemma 7.2.1 (Parallel Transport stays in Ures).Let u(t) , t ∈ I be a (piecewise C1) path in Ures ⊂ GLres and u(t) a horizontal lift of u(t) withinitial conditions u(0) = [U, r] ∈ Ures(H) ⊂ GLres(H).Then u(t) ∈ Ures(H) ∀t ∈ I, i.e. the horizontal lift remains unitary.

Proof. If u(t) = [u(t), q(t)] is the horizontal lift, consider the path

u(t) := [u(t), p(t)], with p(t) := (q∗(t))−1

Clearly, π u(t) = u(t) and u(0) = u(0) = [U, r] ∈ Ures. Furthermore, by (7.1.14), using thefact that u(t) is unitary:

0 = tr[(u(t)u−1(t))++ − q(t)q−1(t)

]= tr

[(u(t)u∗(t))++ − q∗−1(t)q∗(t)

]c.c.= tr

[(u(t)u∗(t))++ − p(t) ˙p−1(t)

]c.c.And using

0 =d

dt

(u(t)u∗(t)

)= u(t)u∗(t) + u(t)u∗(t)

0 =d

dt(p(t)p−1(t)) = p(t)p−1(t) + p(t) ˙p−1(t),

we derivetr[(u(t)u−1(t))++ − p(t)p−1(t)

]= 0.

Thus, u(t) is also a horizontal lift of u(t) satisfying the same initial condition. By uniquenessof the horizontal lift it follows that u(t) ≡ u(t) and thus u(t) ∈ Ures(H) ⊂ GLres(H), ∀t ∈ I.

7.3 Classification of Connections

A connection is always an additional geometric structure on the bundle and as such constitutesa particular choice. We have already restricted this choice to right-inavariant connections. Notethat every connection is by definition invariant under the fibre-preserving right-action of thestructure group on the principle bundle, denoted by rc for c ∈ C×. But our condition is muchstronger: we demand that it is invariant under the right-action of GLres on itself as a Lie group(denoted by a capital R). That is, if γ(t) is a horizontal path in GLres, so is Rg γ(t) = γ(t)· g forany g ∈ GLres. We will see that this assures that the horizontal lift of a unitary time evolutionpreserves the semi-group structure, i.e. is indeed a time evolution on the fermionic Fock space.However, the condition of right-invarience does not specify a unique connection. This is evidentalready from our explicit construction of the Langmann-Mickelsson connection one-form. Itinvolved a projection onto C that was given with respect to an isomorphism g1 → g1 ⊕ C.This isomorphism however is not canonical, but induced from the local trivialization φ or,equivalently, from the local section σ. A different section will define a different isomorphismand consequently a different right-invariant connection.

Generally, every local section υ : GLres → GLres around the identity with υ(e) = e (e beingthe neutral element in the group, i.e. the identity) induces a local trivialization of GL

0

res by

φ−1υ : GL0

res(H)× C× → GL0

res, (g, λ) 7→ υ(g) · λ. (7.3.1)

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The differential Deφυ : Te GLres → Te GLres ⊕ C induces a linear isomorphism

φυ : g1

∼=−−→ g1 ⊕ C, (7.3.2)

which becomes a Lie algebra isomorphism, if the Lie bracket on g1 ⊕C is defined as in (3.5.6)to include the algebra 2-cocycle induced by υ. Then, the canonical right-invarient Maurer-Cartan form, followed by the projection defined by this isomorphism, defines a right-invariantconnection one-form. Conversely, it is true that every right-invariant connection on GLres(H)

comes from a one-form of this kind.

We can understand all this better from a more abstract, geometrical perspective. Rememberthe geometric interpretation of a connection as a distribution in the tangent-bundle TGLres,distinguishing horizontal vectors complementary to ker(Dπ), which is spanned by vector fieldswith flow-lines along the fibres of the principle bundle.Formally, a connection is a smooth subbundle Γ ⊂ TGLres satisfying ∀ p ∈ GLres(H):

i) TpGLres = ker(Dpπ)⊕ Γp

ii) (rc)∗Γp = Γpc, ∀c ∈ C×(7.3.3)

A right-invariant connection has to satisfy in addition4

iii) (Rg)∗Γp = Γpg, ∀g ∈ GLres(H). (7.3.4)

What this tells us is that a right-invariant connection is uniquely determined by the choiceof a subspace Γe complementary to kerDeπ in TeGLres. Sine then, by iii), we have

Γp = (Rp)∗ Γe, ∀p ∈ GLres(H).

We may think of such a subspace Γe as an imbedding of TeGLres∼= g into TeGLres

∼= g. Andsuch an imbedding can always be realized by the differential Deυ : TeGLres → TeGLres of alocal section υ : GLres → GLres. This brings us back to the relationship between local sectionsand bundle connections discussed above for the connection one-forms.

Lemma 7.3.1 (Right-invariant Connections and local Sections).Let Γ be a connection on GLres, invariant under the right-action of the Lie group on itself. Letυ : GLres → GLres be a local section around the identity with υ(e) = e and Deυ(TeGLres) = Γe.Then Γ corresponds to the connection one-form Θυ := prυC ωR.

υ : GLres → GLres

Dυ

oo // φυ : GLres → GLres × C×

Φυ

TpGLres

∼= ker(Dpπ)⊕Deυ(TeGLres)

oo // g ∼= g⊕ C

Γ∣∣p= (Rp)∗Deυ(TeGLres) oo // Θυ = prυC ωR

4This property can actually be regarded as a strenghtened version of ii), although, strictly speaking,the actions involved are not really the same.

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Proof. Note that im(Deυ) = ker(prυC). By assumption, X ∈ TeGLres is horizontal, i.e. X ∈ Γe,if and only if X ∈ im(Deυ). And this holds, if and only if X ∈ ker

(prυC)= ker

(prυC ωR(e)

).

Since both, the distribution Γ and the kernel of prυC ωR are invariant under the right-actionof GLres on itself, the identity holds everywhere.

Theorem 7.3.2 (Uniqueness of the Connection).The connection ΓΘ defined in (7.1.11) is the unique connection on GLres(H) which is invariantunder the right-action of the Lie group on itself and whose curvature equals the Schwingercocycle c in the sense of Prop. 7.1.5. Horizontal lift from g1

∼= TeGLres to g1∼= TeGLres

with respect to this connection, corresponds to the second quantization dΓ, defined by normalordering, in the Fock representation on F =

∧H+ ⊗

∧C(H−).

Proof. Recall that the connection ΓΘ comes from the local trivialization defined by the sectionτ (4.1.5). If we take a connection Γ′, different from ΓΘ but also invariant under the right-actionof GLres, the previous Lemma tells us that it comes from a local section υ with Deυ 6= Deτ .From the discussion of central extensions of Lie algebras in §3.5. we know that this means thatthe Lie algebra cocycle c corresponding to υ differs from the Schwinger cocycle, correspondingto τ , by a homomorphism (a “coboundary” in the sense of cohomology)

µ = Deυ −Deτ = υ − τ : g1 → C

i.e. c(X,Y ) = c(X,Y ) − µ([X,Y ]) for X,Y ∈ g1. Therefore, in the sense of Prop. 7.1.5, thecurvature of the connection Γ′ differs from the Schwinger cocycle by µ([·, ·]).

Finally, comparison with (5.1.32) tells us that under the isomorphism (5.5.1), horizontallift w.r.to the connection ΓΘ corresponds to the second quantization prescription dΓ onF =

∧H+ ⊗

∧C(H−), since in both cases, the resulting cocycle is the Schwinger-term.

In this sense, the connection we have defined is “unique”. Of course, it’s in no way necessary todemand that the curvature of the connection equals the Schwinger cocycle. We can just agreethat it’s nice if it does. Also note that the arguments used in the proof of Thm. 7.3.2, togetherwith the non-triviality of the Schwinger cocycle, prove that there exists no flat right-invariantconnection on GLres(H).

100

Chapter 8

Geometric Second Quantization

Parallel Transport in the GLres(H)-bundle was suggested by E.Langmann and J.Mickelsson asa method to fix the phase of the second quantized scattering matrix in external-field QED.We will call this method Geometric Second Quantization. The bundle connection allows usto lift – in a unique way – paths from the base manifold Ures(H) to the central extensionUres(H) ⊂ GLres(H) that carries the information about the geometric phase. It is importantthat these horizontal lifts are determined by the connection, i.e. the geometric structure ofthe bundle only. We will prove that this fact ensures that the geometric second quantizationis causal, i.e. preserves the causal structure of the one-particle Dirac theory.

We present the method of geometric second quantization in a more general setting, as amethod of second quantization of the whole unitary time evolution. However, the time evolu-tion will always require renormalization and the physical significance of the renormalized timeevolution is unclear. We will discuss this problem in the final chapter of this work.

One might think that the phase of the S-matrix is of little relevance in a quantum theory, orsimply a consequence of the gauge-freedom in QED. But this is not the case. In mathematicallyrigorous formulations of the theory, the phase of the second quantized scattering matrix doesappear as a relevant quantity and it seems to be the ill-definedness of this quantity that leads to(additional) divergences in perturbation theory. It is well known that the vacuum polarizationin QED is ill-defined and requires various renormalizations to be made finite (see [Dys51] for avery “honest” computation). One reason for this is that the current density, usually defined as

jµ(x) = eΨ(x)γµΨ(x), (8.0.1)

is not a well-defined object in the second quantized theory. The better definition can be givenin terms of the second quantized scattering operator S by

jµ(x) := iS∗δ

δAµ(x)S[A] (8.0.2)

which equals e : Ψ(x)γµΨ(x) : in first order perturbation theory (see [Scha], §2.10). Herby,S[A] denotes the map sending A ∈ C∞c (R4,R4) to the second quantized scattering operator inUres(H), corresponding to the interaction defined by A. The current-density itself has to be

101

8.1. RENORMALIZATION OF THE TIME EVOLUTION

understood as an operator-valued distribution. The vacuum-polarization is then well-definedas the vacuum expectation value

⟨Ω, jµ(x)Ω

⟩= i⟨S Ω,

δS

δAµ(x)Ω⟩

or, more precisely, as a distribution evaluated at the test-function A1,

⟨Ω, j[A](A1)Ω

⟩=i

∂

∂ε

∣∣ε=0

⟨Ω, S−1(A)S(A+ εA1)Ω

⟩=i

∂

∂ε

∣∣ε=0

log⟨Ω, S−1(A)S(A+ εA1)Ω

⟩.

Here, the phase of the S-operator enters explicitely. If we separate the phase-freedom, we find

S[A] = S[A]eiϕ[A] ⇒ δS

δAµ(x)= i

δϕ

δAµ(x)S + eiϕ

δS

δAµ(x)(8.0.3)

and for the vacuum-polarization:

⟨Ω, jµ(x)Ω

⟩= i[⟨

Ω, iδ ϕ[A]

δAµ(x)Ω⟩

+⟨S Ω,

δS

δAµ(x)Ω⟩]

(8.0.4)

The second term on the right-hand side is well-defined, but the first term obviously requires awell-defined prescription for the phase of the S-matrix. We will try to provide this now.

8.1 Renormalization of the Time Evolution

Our motivation for studying the parallel transport is the second quantization of the Dirac timeevolution. Given an external field A = (A0,−A) ∈ C∞c (R4,R4), we have the correspondingunitary time evolution UA(t, t′). It is usually more convenient to study the time evolution inthe interaction-picture which is related to UA(t, t′) by

UAI (t, t′) = U0(0, t)UA(t, t′)U0(t′, 0); t, t′ ∈ R. (8.1.1)

A second quantization of the time evolution between t1 and t2 corresponds to a lift of the path

s 7→ UAI (t1 + s, t1), s ∈ [0, t2 − t1] (8.1.2)

to the group Ures(H) that acts on the Fock space. This would provide a well-defined prescriptionfor the implementation of the time evolution on the Fock space, including phase. In particular,since the interaction has time-support contained in some compact interval [−T, T ] for T largeenough,

t 7→ UAI (t,−T ), t ∈ [−T, T ] (8.1.3)

is a path from the identity 1H to the S-Matrix S = UAI (T,−T ) = UA

I (∞,−∞).

Now a bundle connection on Ures(H) or GLres(H), respectively, as introduced in the previouschapter, does precisely that: it defines unique lifts of (smooth) paths from Ures(H) to the prin-ciple bundle Ures(H). However, as the theorem of Ruijsenaars (see Thm. 1.2.1 and especially

102


Thm. 6.1.1) tells us, the paths (8.1.2) are typically NOT in Ures(H); in fact they will leaveUres(H) as soon as the spatial component A of the interaction potential becomes non-zero.Therefore, in order to be able to apply the method of parallel transport, Langmann and Mick-elsson introduced a renormalization of the time evolution, such that the transformed time evo-lution UA

ren(t, t′) stays in Ures(H), for all t, t′ ∈ R and such that UAren(T,−T ) = UA

I (T, T ) = S.

Concretely, they prove the following:

Theorem 8.1.1 (Langmann,Mickelsson 1996).Let A ∈ C∞c (R4,R4) a 4-vector potential and UA(t, t′) the corresponding Dirac time evolution.There is a family of unitary time evolutions Tt(A), t ∈ R such that the modified time evolution

T−1t (A)UA(t, t′) Tt′

belongs to Ures(H) for all t, t′ ∈ R.1

Moreover, T(A) can be chosen such that Tt(A) = 1 if A(t) = 0 and ∂tA(t) = 0.

An explicit expression for the operators Tt(A) is given in [Mi98]. With the abbreviations/A =

∑µ α

µAµ and /E := ∂t/A− /pA0 + [A0, /A] it is defined by

T∗t (A) = U0(t, 0) exp( 1

4

[D−1

0 , /A]− 1

8

[D−1

0/AD−1

0 , /A]− i

4D−1

0/ED−1

0

)U0(0, t). (8.1.4)

In the original paper of Langmann and Mickelsson, the renormalization appears as a meretechnical tool - the meaning of the unitary transformations Tt(A) is not discussed. Apartfrom the question of differentiability, which will be the focus of the next section, this meaningbecomes more evident by the following considerations:

Let U(t, t′) by the unitary evolution for a fixed external field A and let T(t), t ∈ R, a family ofunitary operators such that T(t)−1U(t, t′) T(t′) ∈ U0

res(H), ∀t, t′ ∈ R and T(t) = 1 for |t| largeenough. In particular, for t0 0 outside the time-support of A, this means

T−1(t)U(t, t0)[H−] = [H−] ∈ Pol(H)/ ≈0

⇐⇒ U(t, t0)[H−] = T(t)[H−] ∈ Pol(H)/ ≈0

⇐⇒ U(t, t0) ∈ U0res(H; [H−], [T(t)H−]).

Hence, by the composition property (2.1.4),

U(t, t′) = U(t, t0)U(t0, t′) ∈ U0

res(H; [T(t′)H−], [T(t)H−]). (8.1.5)

Ergo, the operators T(t) identify the correct polarization classes, between which the Dirac timeevolution is mapping. In other words, the renormalization satisfies

[Tt(A)H−]≈0= C[A(t)] ∈ Pol(H)/≈0

, (8.1.6)

where C[A(t)] = C[A(t)] are the polarization classes identified in Thm. 6.1.1.

1Note that the roles of T and T−1 are interchanged in our convention as compared to [LaMi96].

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So we see that we can interpret the renormalization in two different ways:

1. We can use the Tt-operators to renormalize the unitary time evolution, i.e. trans-form it back to Ures(H) and implement it on the standard Fock space. We will seethat a smoothening renormalization is actually equivalent to a renormalization of theHamiltonian, i.e. it results in a modification of the interaction-potential that makes itwell-behaved and prevent the creation of infinitely many particles.

2. We can use the “renormalization” to identify the correct polarization classes and imple-ment the time evolution as unitary transformations between time-varying Fock spaces asexplained in Chapter 6. Obviously, the unitary operators contain even more informationthat we can use to identify instantaneous vacuum states. For the geometric construction,this means that we identify

W (t) := T(t)H− (8.1.7)

as the new projective vacuum at time t and use it to construct the geometric Fockspace FW (t)

geom . In the language of the infinite wedge spaces, we start with a Dirac seaΦ0 : ` → H− ∈ Ocean(H−) corresponding to the free vacuum state in the initial Fockspace FS0

with S0 = S(Φ0). Then, T(t)Φ0 is a Dirac sea with image in the polarizationclass [U(t, t0)H−]≈0 and we can implement U(t, t0) as a unitary map between the Fockspaces FS0

and FSt , where St = S(T(t)Φ0).

The corresponding physical picture is that the Dirac sea is being “rotated” with time,thereby changing our notion of what we call the vacuum, i.e. which configuration of theDirac sea we perceive as “empty”.

With this understanding it becomes clear that such a renormalization is not at all unique butrepresents a very particular choice. It seems that this point was not evident to Langmann andMickelsson by the time of their ’96 publication, although they pick up the issue in a later pub-lication [Mi98]. In [LaMi96], however, it is not discussed whether and how the results dependon the particular choice of the renormalization. Bad news is that, as it turns out, the entirefreedom of the geometric phase is now contained in the freedom of choice of a renormalization.It merely gets a different name: geometrically, it is described by the holonomy group of theprinciple bundle. We will make this more precise in Section 8.3.

To study renormalizations more thoroughly, we propose a general definition:

Definition 8.1.2 (Space of Vector Potentials).

Let A be the space of 4-vector potentials (equipped with a suitable topology).

In our context, A = C∞c (R4,R4) and we write A 3 A = (Aµ)µ=0,1,2,3 = (A0,−A).

On a general space-time R×M , withM a compact manifold without boundary, A correspondsto the space A = Ω1(R×M,R) of smooth connection one-forms.2

By A(t) we always mean the function A(t, ·) ∈ C∞c (R3,R4) for fixed t ∈ R.

2More generally, the one-forms take values in the Lie algebra of the gauge group which for QED isjust Lie(U(1)) = R.

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Definition 8.1.3 (Renormalization).

We call a mapping T : R× A→ U(H) a renormalization if it satisfies

i) Tt(0) ≡ 1

ii) T(t,A) = Tt(A) ∈ U0res

(H; [H−], C[A(t)]

), ∀t ∈ R

iii) Tt(A) depends only on A(t) and ∂kA(t) for k = 0, 1, ..., n and some n ∈ N

We call T a smoothening renormalization, if it has the additional property

iv) For any A ∈ A, the renormalized (interaction picture) time evolution

UAren(t, s) = eitD0 T∗(t)UA(t, s) T(s)e−isD0 (8.1.8)

is continuously differentiable in t w.r.to the differentiable structure of GLres(H).

Note:

• In our definition, the renormalized time evolution is an interaction-picture time evolution.

• i) and iii) together imply that Tt(A) = 1, whenever A vanishes in some time-intervalaround t. In particular, this assures that a renormalization doesn’t alter the S-operatorfor compactly supported interactions.

• iii) formulates a requirement of causality. It states that the renormalization dependsonly on the A-field, locally in time. In particular, if the renormalization depends only onA(t) and not its time-derivatives, it makes sense to regard it as determining one vacuumstate (and therefore one Fock space) over the polarization class C[A] for every A ∈ A.Then we would have a “global” choice, suitable for any Dirac time evolution, which is ofcourse different (and arguably better) than choosing a family of Fock spaces for a fixedtime evolution. We will come back to this in Section 8.1.2.

Examples 8.1.4 (Renormalizations).

• The renormalization T(A) constructed in [LaMi96] is a smoothening renormalization inthe sense of Def. 8.1.3 (with n=1).

• The operators eQA(t)

introduced in §6.1. provide a renormalization (with n=0) which isnot smoothening (cf. §8.1.2).

Remark 8.1.5 (Interpolation Picture).The most obvious way to transform the time evolution back to Ures(HH) is by the timeevolution itself, i.e. to set Tt(A) = UA(t,−∞). Then, the renormalized time evolution is

U∗I (t,−∞)UI(t, t′)UI(t

′,−∞) = 1, ∀t, t′ ∈ R,

where, t = −∞ can be understood as a large negative t outside the time-support of theinteraction. This approach is also known as “interpolation picture”. The only problem withthe interpolation picture: nothing’s happening. The polarization, i.e what we call “particles”and “antiparticles”, evolves in just the same way as the states themselves:

H = U(t,−∞)H+ ⊕ U(t,−∞)H−. (8.1.9)

105


What we end up caling the “vacuum” at time t is exactly the state into which the original(t → −∞) vacuum has evolved. There is no particle creation or annihilation - an emptyuniverse remains empty. For the obvious reasons, we are not satisfied with that. From theorem6.1.2, we know that at least the S-matrix is in Ures(H), hence we can make sense of theparticle/antiparticle - picture at least asymptotically. Before the interaction is switched onand after it’s switched off, we can determine the particle content with respect to the samevacuum and the question: ‘how many particles and anti-particles were created?’ has a well-defined answer. We demand of our renormalization to grant us this, at least.

8.1.1 Renormalization of the Hamiltonians

Now we shift our focus to the question of differentiability which makes all the difference be-tween a renormalization and a smoothening renormalization. To make use of the geometricstructure introduced in Chapter 7 and apply the method of parallel transport, we need the timeevolution to be differentiable - notably with respect to the differentiable structure on GLres(H)

(or Ures(H), respectively), induced by the norm (2.2.5) on the Banach-algebra Bε(H). Thisnotion of differentiability is very strong. The interaction picture time evolution is generallydifferentiable in the operator norm, but now we need to control the Hilbert-Schmidt norms aswell. We will see that this requirement is not harmless.

Renormalization of the Interaction Hamiltonians

Recall that if UA(t, t′) solves the equation of motioni ∂t U

A(t, t′) = HA(t)UA(t, t′)

UA(t′, t′) = 1

(8.1.10)

with the Hamiltonian

HA = D0 + e

3∑µ=0

αµAµ = D0 + V A(t),

the corresponding interaction-picture time evolution

UAI (t, t′) = eitD0UA(t, t′)e−it

′D0

is a solution to the equivalent equationi ∂t U

AI (t, t′) = HI(t)U

AI (t, t′)

UAI (t′, t′) = 1

(8.1.11)

withHI(t) = eitD0V A(t)e−itD0 . (8.1.12)

This is called the interaction picture. The interaction picture is a hybrid between the Schrödinger-and Heisenberg-picture. The basic idea is that in the interaction-picture, operators evolve ac-cording to the free time evolution. The evolution of the states on the other, is then generatedby the interaction-part of the Hamiltonian only. One advantage of this procedure is that VI(t),in contrast to HA(t), is a bounded operator and thus the solution of (8.1.11) is given for all

106


finite times by the norm-convergent Dyson series, defined by

UAI (t, t′) =

∞∑n=0

Un(t, t′),

U0(t, t′) ≡ 1, Un+1(t, t′) = −it∫

t′

HI(s)Un(s, t′)ds.

(8.1.13)

Now let Tt = Tt(A) be a smoothening renormalization for this time evolution and consider themodified (Schrödinger-picture) time evolution

U ′(t, t′) = T∗t UA(t, t′) Tt′ = e−itD0UA

ren(t, t′)eit′D0 .

Differentiation with respect to t yields:

i ∂t U′(t, t′) = i (∂t T∗t )U

′(t, t′) Tt′ + T∗t HA(t)U ′(t, t′) Tt′

=[i (∂t T∗t ) Tt + T∗t H

A(t) Tt]

T∗t U′(t, t′) Tt′

=[−i T∗t (∂t Tt) + T∗t H

A(t) Tt]U ′(t, t′).

We define [−i T∗t (∂t Tt) + T∗t H

A(t) Tt

]=:(D0 + iV A

ren

)with

V Aren =

[T∗t V

A Tt + T∗t [D0,Tt]− iT∗t (∂t Tt)]. (8.1.14)

Thus the renormalized time evolution is generated by the Hamiltonian HAren = D0 + V A

ren

with the “renormalized” interaction (8.1.14).

Theorem 8.1.6 (Generators of renormalized Time Evolution).Let V (t) = V A

ren(t) be a (renormalized) interaction potential and h(t) = eitD0V (t)e−itD0 .Let U(t, t′)(= UA

ren(t, t′)) be a solution ofi ∂t U(t, t′) = h(t)U(t, t′)

U(t′, t′) = 1

in the operator-norm on U(H), given by the Dyson-series (8.1.13).

i) If U(t, t′) is a solution in Ures(H), i.e. a solution w.r.to the differentiable structure inducedby the norm ‖·‖ε, then

[ε, V (t)] ∈ I2(H), ∀t ∈ R,

i.e. V (t) and h(t) are in the Lie algebra ures of Ures(H).

ii) Conversely, if [ε, V (t)] ∈ I2(H)∀t ∈ R, then U(t, t′) is a solution in Ures(H) ⊂ GLres(H),if additionally we assume ∫

R‖[ε, V (t)]‖2 dt <∞. (8.1.15)

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What is actually proven in [LaMi96] for the renormalization constructed by Langmann andMickelsson is that the renormalized interaction stays in ures. It follows that their unitarytransformations are indeed smoothening renormalization in the sense of our Definition 8.1.3.

Note that for U(t, t′) ∈ Ures(H) it would suffice that

t∫t′

[ε , h(t)

]dt =

t∫t′

[ε , eiD0tV (t)e−iD0t

]dt <∞ ∀t, t′ (8.1.16)

which is much less restrictive than (8.1.15).

The proof of the theorem requires some estimates.

Lemma 8.1.7 (Estimates).For the terms in the Dyson series (8.1.13) we get the norm-estimates

‖Un(t, t′)‖ ≤ 1

n!

( t∫t′

‖V (s)‖ds

)n, ∀n ≥ 0 (8.1.17)

and

‖[ε, U1(t, t′)]‖2 ≤t∫

t′

‖[ε, V (s)]‖2 ds

‖[ε, Un(t, t′)]‖2 ≤1

(n− 2)!

t∫t′

‖[ε, V (s)]‖2 ds

( t∫t′

‖V (r)‖dr

)n−1

, ∀n ≥ 1

(8.1.18)

Proof. See Appendix A.3.

Proof of the Theorem. First we note that V (t) is Hermitian with [ε, V (t)] ∈ I2(H) if and only ifh(t) = HI(t) = eitD0V (t)e−itD0 is, because eitD0 is unitary and diagonal w.r.to the polarizationH = H+ ⊕ H−. Now suppose U(t, t′) is in Ures(H) for all t, t′ and differentiable in the norm‖·‖ε, solving i ∂t U(t, t′) = h(t)U(t, t′). Then, for any fixed t:

−i h(t) =d

ds

∣∣∣s=0

U(t+ s, t′)U∗(t, t′) ∈ TeUres∼= (−i) · ures.

Conversely, if∫R‖[ε, V (t)]‖2 dt < ∞, the Hilbert-Schmidt norm estimates in the previous

Lemma show that U(t, t′) ∈ Ures(H), ∀t, t′ and

U(t, t′) = 1− i∫ t

t′h(s)ds+O(|t− t′|2),

so that indeed i ∂t U(t, t′) = h(t)U(t, t′) (as long as s 7→ h(s) is continuous in ures).

108


Corollary 8.1.8 (Time Evolution always requires renormalization).Let A = (Aµ)µ=0,1,2,3 = (Φ,−A) ∈ A. The interaction potential

V A(t) = e αµAµ = −e α ·A+ eΦ (8.1.19)

is not in ures, unless A ≡ 0 and ∇Φ ≡ 0. Consequently, UAI (t, t′) is not differentiable in the

‖·‖ε-norm, except for those cases.

Remark: In general, only the weaker condition

t∫t′

[ε, eiD0tΦ(t, x)e−iD0t] <∞ (8.1.20)

is satisfied ([Ruij77]), which implies UΦ(t, t′) ∈ Ures(H) but not the required differentiability.

Proof. We apply Theorem 8.1.6: We know that UAI (t, t′) is not even in Ures(H), unless A ≡ 0,

and hence V A(t) /∈ ures, unless A ≡ 0. Now consider a purely electric potential V (t) = eΦ,where Φ(x) = Φ(x)·1C4 has to be understood as the multiplication operator in L2(R3,C4) = H.Naturally:

Φ ∈ ures ⇐⇒ exp(iΦ) ∈ Ures(H).

But we know when the latter is the case. eiΦ(x) is a gauge transformation and thus Thm. 6.2.2tells us that it is in Ures(H) if and only if ∇Φ(t, x) ≡ 0. This finishes the proof.

This observation reveals somewhat of a shortcoming of the method of parallel transport: Evenfor purely electric potentials, when the time evolution actually stays in Ures(H), i.e is imple-mentable on the standard Fock space, it requires us to apply a smoothening renormalization,in order to get a differentiable path in Ures(H).

8.1.2 Notes on Renormalizations

We have proposed the general definition 8.1.3 of a “renormalization” in order to establish a com-mon framework for discussing and comparing the constructions of Langmann and Mickelsson[LaMi96] and Deckert et.al. [DeDuMeScho]. We have also explained how such a renormal-ization can be used to translate between the second quantization of the renormalized timeevolution and the implementation on time-varying Fock spaces (see also the proof of Thm.8.2.4 for an actual application of this duality). However, our analysis also shows in what sensethis treatment is justified and when we have to differentiate. In particular cases we will haveto be very careful with the way the renormalization is actually used.

For example, we need a smoothening renormalization for second quantization of the Hamil-tonians, or for applying the method of parallel transport. If however we want to translate theresults into the language of time-varying Fock spaces and use the same renormalization toidentify instantaneous Fock spaces (respectively vacua), these choices will be restricted by thecondition that the renormalized time evolution be differentiable in Ures(H) which is not a sen-sible requirement any more in the context of time-varying Fock spaces. In particular, the lastcorollary tells us that we might have to change vacua/Fock spaces, even if the time evolutionactually stays in Ures(H).

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On the other hand, if we are mainly interested in renormalizations as a mean to identify thecorrect polarization classes and pick out instantaneous vacua/Fock spaces, we have been verygenerous with the kind of choices that we allow, because we know that the polarization classesactually depend on the spatial part of the electromagnetic potential at fixed time t, only, i.eC(t) = C[A(t)]. In this context, it could make more sense to use a “renormalization” whichdepends only on the spatial part of the A-field, locally in time. This would correspond toa “global” choice of Fock spaces, respectively vacua, rather than allowing different choice forevery single time evolution. One could call such a renormalization minimal, because it requiresonly the minimal amount of information from the A-field. We formalize this in the followingdefinition:

Definition 8.1.9 (Minimal Renormalization).Let A3 = C∞c (R3,R3) (or A3 = Ω1(M,R)) be the space of static, space-like vector potentials.We call a map T : A3 → U(H) satisfying

i) T(0) = 1

ii) T(A) ∈ U0res

(H; [H−], C[A]

), ∀A ∈ A3

a minimal renormalization or a global choice of vacua.

We can regard the set of polarization classes C[A] as a formal U0res-bundle over A3. A minimal

renormalization T would then correspond to a global section in this bundle, determining onevacuum – and thereby one geometric Fock space – over every polarization class C[A]. Similarly,for the infinite wedge space construction, if we have chosen a sea Φ0 ∈ Ocean(H−) correspond-ing to the free vacuum state, the assignment A 7−→ S

(T(A)Φ0

)defines a global choice of Dirac

sea classes, and thereby a global choice of infinite wedge spaces over the corresponding polar-ization classes. For a mathematical treatment of the Fock space bundle see [CaMiMu00] and[CaMiMu97], for example. This quite abstract construction however remains to be comparedand reconciled with the constructions and results in the present work.

Examples 8.1.10 (Minimal Renormalizations).

1. For A ∈ A3, we can set A = (0,−A) ∈ C∞c (R3,R4) and T(A) := eQA

. By Thm. 6.1.1[Identification of Polarization classes], this defines a minimal renormalization in the senseof the previous definition.

2. For A ∈ A3, we can take T(A) to be any unitary transformation sending H− to PA−(H),

where PA± are the orthogonal projectors corresponding to the spectral decomposition

w.r.to the Hamiltonian with static vector potential A = (0,−A). By Thm. 6.2.1 [Scharf,Fierz], this defines a minimal renormalization in the sense of the previous definition. Theresulting choice of instantaneous vacua is also known as the Furry picture.

Is it possible to define a smoothening renormalization which is “minimal” in this sense?The answer is no. And the reason is Corollary 8.1.8, saying that even for a purely electricpotential, the time evolution requires renormalization to become differentiable in GLres(H). Isit possible, at least, to find a smoothening renormalization which depends only on the A-fielditself, locally in time, and not on any time-derivatives? Again, the answer is no, as we willshow in the following proposition.

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Proposition 8.1.11 (Strictly Causal Renormalizations).Let T : A → U(H) be a renormalization, such that for every fixed t ∈ R, Tt(A) is only afunction of A(t) ∈ C∞c (R3,R4). Then, T is not smoothening.

Proof. Given A ∈ A, we may write Tt := Tt(A) = T(A(t)), since Tt depends only on A(t).Suppose T was in fact a smoothening renormalization. Then we know from Thm. 8.1.6[Generators of the renormalized time evolution] and the differential form of the renormalization(8.1.14), that [

T∗t VA(t) Tt + T∗t [D0,Tt]− iT∗t (∂t Tt)

]∈ ures (8.1.21)

holds true for all A ∈ A and all t ∈ R. But for any fixed t, we may use this identity for avector potential A′ ∈ A which is constantly equal to A(t) in a time interval around t, i.e. whichsatisfies A′(s) = A(t), ∀ s ∈ (t−ε, t+ε). For the field A′ then, the last term in (8.1.21) vanishes,whereas the first two terms must agree with those for A, because Tt depends only on A(t). Weconclude that (8.1.21) implies that both

a) T∗(A(t))V A(t) T(A(t)) + T∗(A(t))[D0,T(A(t))] ∈ ures

and b) − i T∗(A(t)) ∂t T(A(t)) ∈ ures

hold separately for any fixed t ∈ R. Writing T(A) = eQ(A) for all static fields A ∈ C∞c (R3,R4),with anti-Hermitian operators Q(A) (not the ones defined in §6.1!), property b) states thatt → Q(A(t)) is differentiable for all A ∈ A with [ε, Q(A(t))] ∈ I2(H). But this cannot be thecase. If we take, for example, an A ∈ C∞c (R4,R4) with A(t) ≡ 0 for t ≤ 0, but A(1) 6= 0, weconclude

∥∥[ε,Q(A(1))]∥∥

2=∥∥[ε, 1∫

0

Q(A(t)) dt]∥∥

2≤

1∫0

∥∥[ε, Q(A(t))]∥∥

2dt <∞

(assuming integrability on the right-hand side) and thus Q(A(1)) ∈ ures, hence T1(A) =

eQ(A(1)) ∈ Ures(H). This is in contradiction with T being a renormalization, because A(1) 6= 0

implies by Thm. 6.1.1 : C[A(t)] 6= [H−]. We conclude that the renormalization T cannot besmoothening.

These considerations help to illustrate how restrictive the “smoothing” property for a renor-malization actually is. A smoothening renormalization will always require more informationon the interaction potential, than is necessary to identify the polarization classes, because ithas to control the infinitesimal variation of the field as well. In particular, it follows that therenormalization defined by the operators eQ

A(t)

, constructed in [DeDuMeScho] and introducedin Section 6.1, is not smoothening.

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8.2. GEOMETRIC SECOND QUANTIZATION

8.2 Geometric Second Quantization

Finally, we define the method of second quantization by parallel transport in the principlebundle GLres(H)→ GLres(H), w.r.to the connection ΓΘ defined by the one-form (7.1.11).

In the following, let A ∈ C∞c (R4,R4) and UAI (t, t′), t, t′,∈ R ∪ ±∞ the unitary (interaction

picture) time evolution for the external field A. After applying a suitable smoothening renor-malization Tt as definded in the pevious section, the renormalized time evolution UA

ren(t, t′) isa two-parameter semi-group in Ures(H) ⊂ GLres(H) and continuously differentiable in t withrespect to the differentiable structure on GLres(H). We construct its second quantization i.e.the lift to the group Ures(H) acting on the Fock space by the following prescription:

For t1 ≥ t0 ∈ R we define the time evolution

U(t, t0), t ∈ [t0, t1] (8.2.1)

between t1 and t0 on the Fock space as the ΓΘ-horizontal liftof the renormalized one-particle time evolution UA

ren(t, t0), t ∈ [t0, t1]

to Ures(H) with initial condition U(t0, t0) = 1Ures.

In other words: for t1 ≥ t0 ∈ R we define the lift U(t1, t0) ∈ Ures(H) of UAren(t1, t0) as the

parallel transport of 1 ∈ Ures(H) along the path

s→ UAren(t0 + s, t0), s ∈ [0, t1 − t0]

in Ures(H) with respect to the connection determined by Θ.

This definition yields a well-defined lift of the renormalized time evolution to the fermionicFock space. In particular, it gives a smooth prescription for the phase of the implementations.

Figure 8.1: Lifting the time evolution by parallel transport.

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Second Quantization of the S-matrix

Since A has compact support in time, so do UAI (·, ·) and the renormalization Tt(A). Therefore,

there exists T > 0 such that UAren(t1, t0) = UA

I (T,−T ), whenever t1 ≥ T, t0 ≤ −T .In particular, for any such T ∈ R:

UAren(T,−T ) = UA

I (∞,−∞) = S

and the S-matrix is unaltered by the renormalization. Therefore,

γ : s −→ UAren(−T + s,−T ), s ∈ [0, 2T ] (8.2.2)

is a differentiable path from 1 to S in Ures(H) ⊂ GLres(H). The second quantization S of thescattering operator S is then defined as the parallel transport of 1 ∈ Ures(H) along γ.

This procedure determines the second quantized scattering matrix in a well-defined manner.However, as we will see, the phase of S will depend on the choice of the renormalization.Therefore, it makes sense to write

S = S[A,T]; A ∈ A,T a renormalization. (8.2.3)

Second Quantization of the generators

A connection on the principle bundle allows us to lift not only paths, but also vector fieldsfrom the base-manifold to the principle bundle in a unique way. In our setting, this meansthat it distinguishes unique lifts of the renormalized interaction Hamiltonians to the universalcovering of the Lie algebra. One might call this a second quantization of the Hamiltonians.

Theorem 8.2.1 (Second Quantization of the Hamiltonians).

i) Horizontal lifts with respect to the connection ΓΘ yields a continuous section

dΓ : ures → ures (8.2.4)

of the Lie algebra ures of Ures(H) into its central extension ures. This defines a “secondquantization” of self-adjoint operators satisfying the Shale-Stinespring condition.The section gives rise to a commutator anomaly

[dΓ(X),dΓ(Y )] = c(X,Y ) dΓ([X,Y ]) (8.2.5)

with c ∈ H2(ures,R) the Schwinger-cocylce (4.1.10).

ii) If V Aren is the renormalized interaction (8.1.14) and

h(t) = h[A, T ](t) := eiD0tV Aren(t)e−iD0t (8.2.6)

the renormalized interaction Hamiltonian, the so defined second quantizationh(t) := idΓ(−i h(t)) is the generator of the time evolution U(t, t′) on the Fock space.

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Diagrammatically, if Γ denotes second quantization of unitary operators and dΓ second quan-tization of their self-adjoint generators, we have

ures h(t)

π

exp // U(t, t′)

π

Ures(H)

ures h(t)

dΓ

OO

exp // UA(t, t′)

Γ

OO

Ures(H)

What is essential here (and not expressed in the diagram) is that the lift of the Hamiltoniansare determined by the geometric structure alone.

Proof of the Theorem.

i) A connection in general and ΓΘ in particular, determines unique horizontal lifts of vectorson the base manifold Ures(H) to the principle bundle Ures(H).Under the canonical identification

ures∼= Te Ures, ures

∼= Te Ures

this yields the desired section dΓ in the central extension of Lie algebras:

0 // Rı// ures

π// ures

dΓ

||// 0 . (8.2.7)

We know from the discussion in Section 3.5 that such a section gives rise to a Lie algebra2- cocycle and it follows from Prop. 7.1.5 that the cocycle corresponding to dΓ equals theSchwinger term.

ii) h(t) is defined by −i h(t) = dΓ(−ih(t)) for every t ∈ R.For fixed t0, the renormalized time evolution UA

ren(t, t′) satisfies

i ∂t UAren(t, t0) = h(t)UA

ren(t, t0).

By construction, t 7→ U(t, t0), t ∈ [t0,∞) is precisely the integral curve to the horizontallift of the vector field

Xγ(t) := −i(RU(t,t0)

)∗ h(t) = −i h(t)U(t, t0) (8.2.8)

along γ(t) = UAren(t, t0), t ∈ [t0,∞). Since the connection ΓΘ is invariant under the

right-action of Ures(H) on itself,

Xγ(t) := −i(RU(t,t′)

)∗ h(t) = −i h(t)U(t, t′)

is horizontal for all t ∈ R and obviously a lift of Xγ(t) to the tangent bundle T Ures ofUres(H). It follows that U(t, t′) satisfies

i ∂t U(t, t0) = h(t)U(t, t0), ∀t, t0 ∈ R

hence h(t) generates the second quantized time evolution U(t, t0), as was claimed.

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8.2.1 Causality

For the second quantized time evolution U(t, t′) we can derive the following important result:

Theorem 8.2.2 (Semigroup Structure of the Time Evolution).The lifted time evolution U(t, t′) in Ures(H), defined by horizontal lifts as above, preserves thesemi-group structure of the time evolution i.e. satisfies

U(t, t) = 1Ures∀ t ∈ R

U(t2, t1)U(t1, t0) = U(t2, t0) ∀ t0 ≤ t1 ≤ t2 ∈ R(8.2.9)

These properties justify its denomination as a “time evolution” on the Fock space.

Proof. U(t, t) = 1,∀ t ∈ R holds by construction. For the composition property, note thatU(t2, t0) is the end-point of the horizontal lift of

s→ UAren(t1 + s, t0), s ∈ [0, t2 − t1]

with starting point U(t1, t0). On the other hand, consider the curve

s→ U(s, t1)U(t1, t0), s ∈ [0, t2 − t1]

It has the same starting point U(t1, t0) ∈ Ures(H) and projects down to

π(U(s, t1)U(t1, t0)) = UA

ren(s, t1)UAren(t1, t0) = UA

ren(s, t0) ∈ Ures(H).

Furthermore, dds U(s, t1)U(t1, t0) = (RU(t1,t0))∗ U(s, t1) is horizontal, because X(s) := U(s, t1) is

horizontal by construction of the lift and the connection ΓΘ is invariant under right-action ofUres on itself. That is, if Θ denotes the connection one-form, we have:

Θ((RU(t1,t0))∗X(s)

)= (RU(t1,t0))

∗Θ (X(s)) = Θ(X(s)) = 0

By uniqueness of the parallel transport it follows that both curves are actually the same.In particular U(t2, t1)U(t1, t0) = U(t2, t0).

Note that the same argument wouldn’t go through with the left-invariant Langmann-Mickelssonconnection. We could however obtain the analogous result by lifting the unitary evolution“backwards” in time, i.e. by lifting the path

s→ UAren(t1, t1 − s), s ∈ [0, t1 − t0]

This is a perfectly valid procedure, but seems more artificial from a physicist’s point of view.Therefore we proposed the right-invariant version of the Langmann-Mickelsson connection; thisconvention fits better with the usual form of the time evolution where subsequent time-stepscorrespond to unitary operators multiplied from the left.

To appreciate the value of this result, let’s first make clear what’s not the point. Thesemi-group structure alone wouldn’t be worth the trouble, it actually comes fairly cheap:

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Proposition 8.2.3 (Lifting the semi-group Structure).Let U(t, t′), t, t′ ∈ R a two-parameter semi-group in Ures(H) with compact support in time.

For every t ∈ R choose any lift U(t,−∞) of U(t,−∞) to Ures(H) and set

U(t1, t0) := U(t1,−∞) U(t0,−∞)−1

for t1 ≥ t0 ∈ R ∪ ±∞. The so defined lift has the semi-group properties (8.2.9).If t→ U(t,−∞) is continuous/differentiable, then U(t, t′) is continuous/differentiable in t.

Proof. For t2 ≥ t1 ≥ t0 ∈ R we find

U(t2, t1) U(t1, t0) = U(t2,−∞) U(t1,−∞)−1 U(t1 −∞) U(t0,−∞)−1

= U(t2,−∞) U(t0,−∞)−1 = U(t2, t0)

And of course U(t, t) = U(t,−∞) U(t,−∞)−1 = 1,∀t ∈ R.

To appreciate the virtue of the geometric construction of the time evolution, we need to under-stand the difference between the results in theorem 8.2.2 and the previous proposition 8.2.3.If the semi-group structure is not the point, then what is?

The composition property

U(t2, t1) U(t1, t0) = U(t2, t0),∀t0 < t1 < t2 ∈ R (8.2.10)

is often related to the physical principle of causality. But at this point, this is actually toobig a word. The algebraic identities by themselves merely express the basic requirements for atwo-parameter family of unitary operators to deserve the title of a “time evolution” in the firstplace. We must arrive at exactly the same state, whether we follow the evolution of a physicalsystem from time t0 to time t1 > t0 and then from time t1 to t2 > t1, or whether we skip theintermediate time-step and follow the evolution from t0 to t2 directly. If what the theory callsa “state” involves a phase, then these phases need to agree as well. So far, this is a questionof consistency rather than causality. To relate the semi-group structure to something like acausal structure of the physical theory, we have to take a closer look at the construction of the(second quantized) time evolution and what it involves.

The construction in Prop. 8.2.3 is not a serious proposal for a second quantization, butrather an instructive example to demonstrate why the semi-group properties (8.2.9) alone areinsufficient. The two-parameter group U(t1, t0) in Ures(H) was constructed by choosing lifts ofthe one-particle time evolution operators U(t,−∞) first and using those lifts to generate theentire family. This has the following effect: if we want to call U(t1, t0) a (second quantized)time evolution, we find that the evolution between times t0 and t1 > t0 depends on the entirehistory of the physical system, i.e. on A(t) for t ∈ (−∞, t1]. If we altered the electromagneticfields in the distant past t t0, we would most likely get a different lift for the time evolutionbetween t0 and t1. It’s debatable whether this should be called a violation of causality (andthe answer will depend on our very definition of that concept). But certainly, such a solutionwould be at odds with our conventional understanding of a deterministic evolution. We would

116


sacrifice what one might call the “Cauchy property”3 of the system: that its future evolutionis completely determined by the laws of physics, given the current state of the system at anymoment in time (or on a spacelike Cauchy-surface in a relativistic setting).

If in the construction of Prop. 8.2.3 we used a different “reference time” than t = −∞(which we could do), the situation would be even worse. The time evolution of the systembefore that particular reference time would then depend on the electromagnetic potential inthe future. Obviously, this constitutes a violation of causality in a very strong sense.4

In our geometric construction, the second quantization of the renormalized time evolutionU(t, t′) and its generators h(t) depend only on the corresponding (“first quantized”) objects inthe one-particle theory, on the renormalization used to transform them back to Ures or ures,respectively, and finally on the bundle-connection introduced in Chapter 7. In particular, thelifts of the renormalized time evolution and the renormalized interaction Hamiltonians aredetermined solely by the geometric structure of the Ures(H) bundle. We have required that therenormalization is causal in the sense that Tt(A) depends only on A(t) and its time-derivativesup to finite order n. Consequently, the geometric second quantization assures that U(t1, t0)

depends on the external field A(t) (and its first n time-derivatives) only at times t ∈ [t0, t1].Similarly, for fixed t ∈ R, the lifted interaction Hamiltonian h(t) depends only on A(t) and itsfirst n+1 time-derivatives at time t. Conclusively, we assert that geometric second quantizationpreserves the causal structure of the one-particle theory.

Application to time-varying Fock spaces

We can translate these insights back into the language of time-varying Fock spaces and derivethe following result:

Theorem 8.2.4 (Time Evolution on time-varying Fock Spaces).Given a unitary time evolution UA(t, t′) and a family of infinite wedge spaces (Ft)t∈R overOcean(C(t)), there exists right-operations by R(t, t′) ∈ U(`) such that

U(t, t′) := LU(t,t′)RR(t,t′) : Ft −→ Ft′ , ∀t, t′ ∈ R

defines a time evolution, i.e satisfies:U(t, t) = IdFt ∀ t ∈ R

U(t2, t1) U(t1, t0) = U(t2, t0), ∀ t0 ≤ t1 ≤ t2 ∈ R(8.2.11)

Moreover, for all t ≥ t′ ∈ R, the operator R(t.t′) depends on the A-field (and possibly its timederivates) only in the time-interval [t′, t].

3Or “Markov property”, if we borrow the language of probability theory.4Of course, we shouldn’t forget that in all this, we are merely talking about a phfibre bundlease-

factor which is not measurable per se. Still, the relative phase between two states or the variation ofthe phase with the electromagnetic potential can (theoretically) have observable consequences (recall(8.0.4), for instance). What this means in practice (in the context of QED), however exceeds theauthor’s competence.

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Proof. Let (Tt)t be a smoothening renormalization for the time evolution UA(t, t′) and letU(t, t′) be the second quantized time evolution on Fgeom constructed in Thm. 8.2.2. We fix aΦ0 ∈ Seas(H) with im(Φ0) = H− so that

∧Φ0 in FS(Φ0) will play the role of the free vacuum

state. Let C(t) = C[A(t)] be the polarization classes identified in Thm. 6.1.1 [Identification ofPolarization classes]. Then we have Tt Φ0 ∈ Ocean(C(t)), ∀t ∈ R and Gt := FS(TtΦ0) definesa family of Fock spaces over C(t), t ∈ R. Under the isomorphism

∧U

defined in Cor. 5.4.3,U(t, t′) becomes a time evolution on FS(Φ0). For every t ≥ t′ ∈ R, there exists an operatorS(t, t′) ∈ U(`) such that:∧

UU(t, t′) = L(UA

ren(t,t′))RS(t,t′) = L(U0(0,t) T∗t UA(t,t′) Tt′ U

0(t,0))RS(t,t′)

andS(t1, t0)S(t2, t1) = S(t2, t0), ∀ t0 ≤ t1 ≤ t2.

We may also choose them such that S(t, t) = 1` for all t. Note that the essential part is thatthose right-operations fit together in the right way to form a two-parameter semi-group. Thus,setting V (t, t′) := LUA(t,t′)RS(t,t′) we get a two-parameter family of unitary transformationsV (t, t′) : Gt′ → Gt with V (t2, t1) V (t1, t0) = V (t2, t0), ∀ t0 < t1 < t2. This, however, isnot really satisfying, yet, because the Fock spaces themselves are chosen by the smootheningrenormalization. If we have chosen a different family (Ft)t∈R of infinite wedge spaces over thecorrect polarization classes, it follows from Cor. 5.2.10 [U(`) acts transitively on oceans] thatthere exists for every t a unitary transformation r(t) ∈ U(`) with Rr(t) : Gt

∼=−−→ Ft. For anyt ≥ t′ ∈ R, we set

R(t, t′) := r∗(t′)S(t, t′)r(t).

Then,U(t, t′) := LU(t,t′)RR(t,t′) = Rr(t)V (t, t′)Rr∗(t′) : F ′t −→ Ft

and for all t0 ≤ t1 ≤ t2 we find

R(t1, t0)R(t2, t1) = r∗(t0)S(t1, t0)r(t1) r∗(t1)S(t2, t1)r(t2)

= r∗(t0)S(t2, t0)r(t2) = R(t2, t0)

so thatU(t2, t1) U(t1, t0) = U(t2, t0).

Also, R(t, t) = r∗(t)S(t, t)r(t) = r∗(t)r(t) = 1`, so that U(t, t) = IdFt , ∀t ∈ R.

Note that the parallel transport applied to time-varying Fock spaces in this way does notanymore eliminate the U(1)-freedom of the lifts. The reason is that the operators r(t) usedin the proof to map the family of Fock spaces (Gt)t (determined by the renormalization) tothe Fock spaces (Ft)t are not unique. If we choose a different family r′(t), they will differ byr′r∗(t) ∈ U1(`), i.e. Rr′(t) = det(r′r∗(t)) · Rr(t) : Gt

∼=−−→ Ft. (See also Cor. 5.2.11 [Equivalenceof infinite wedge spaces]). Thus, the phase-freedom now comes from the ambiguity in identi-fying different, equivalent Fock space representations over the same polarization class.

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The Causal Phase of the S-Matrix

As mentioned before, it is unclear whether the time evolution is meaningful at all in QuantumElectrodynamics. The predominant position seems to be that only the S-matrix is physicallysignificant. If one adheres to this view, the causal properties of the second quantized (renor-malized) time evolution are of secondary interest. A better formulation of causality in termsof the second quantized S-matrix is therefore the following:

Given an external fieldA = A1 + A2 ∈ A (8.2.12)

which splits in two parts with disjoint supports in time, i.e. ∃ r ∈ R such that

suppt A1 ⊂ (−∞, r) , suppt A2 ⊂ (r,+∞). (8.2.13)

That is, the field A1 vanishes for times t ≥ r and the field A2 vanishes for t ≤ r.We say that the phase of the S-matrix is causal if

S[A] = S[A1 + A2] = S[A2] S[A1]. (8.2.14)

The importance of this causality condition was emphasized in particular by G.Scharf in [Scha].But indeed, the causal properties of the geometric second quantization as discussed above doimply causality in the sense of Scharf: As the A-field vanishes around time r, so does therenormalization. Therefore:

UAren(r,−∞) = UA

I (r,−∞) = S[A1]

UAren(+∞, r) = UA

I (+∞, r) = S[A2]

Hence, by construction of the lift and theorem 8.2.2:

S[A] = U(+∞,−∞) = U(+∞, r)U(r,−∞) = S[A2] S[A1]. (8.2.15)

We note that this finding seems to contradict the results in [Scha], where it is suggested thatthe phase of the second quantized scattering operator is completely determined by the causalitycondition (8.2.14). The geometric second quantization of Langmann and Mickelsson is causal,still it does not yield a unique phase (unfortunately). In fact, there is pretty much freedomleft. We can choose a different GLres-invariant connection and/or different renormalizationsof the time evolution. Different connections will inevitable lead to different causal phases, ingeneral, since a bundle connection is uniquely determined by its parallel transport maps. Howthe choice of the renormalization can affect the phase of S is discussed in the next section.

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8.3. HOLONOMY OF THE BUNDLES

8.3 Holonomy of the Bundles

We have seen that the choice of the renormalization is not unique. Therefore, the renormalizedtime evolution is – without additional requirements – very arbitrary and the urging questionarises, how meaningful it can actually be in a physical description. The S-matrix, however,is invariant under renormalizations and one might hope that geometric second quantizationyields a well-defined scattering operator on the Fock space, as was suggested in [LaMi96].Unfortunately, the ambiguity in the renormalization is problematic for the second quantizationof the S-matrix as well. If we use two different renormalizations to implement the S-matrixon the Fock space via parallel transport, we have two different curves in Ures(H) ⊂ GLres(H),both with starting-point 1H ∈ Ures and end-point S ∈ Ures. Consequently, their horizontallifts are two different curves in Ures(H). The starting-points of the lifted paths in Ures(H)

are, by definition, the same (namely 1 ∈ Ures), but their end-points – the corresponding liftof the scattering matrix – need not be. (We just know that they both lie in the same fiberabove Ures(H) 3 π−1(S) ∼= U(1)). Naturally, the question arises how these end-points, i.e. thephase of the scattering matrix, depend on the choice of the renormalization. Geometrically,this freedom is expressed by the holonomy group of the bundle π : Ures(H)→ Ures(H).

Figure 8.2: PT along different paths can end in different points in the fibre over S

Consider a principle-G-bundle Eπ−→ M for a (finite-dimensional) Lie group G over a

connected, paracompact manifold M , equipped with a connection Γ. Let u0, u1 two points onthe base-manifold M and c, c′ : [0, 1] → M two piecewise differentiable curves from u0 to u1 .Let p ∈ π−1(u0) ⊂ E be an element in the fibre over the starting point u0. Now we want toknow if parallel transport of p along c yields the same result as parallel transport of p along theother path c′. In other words, we want to now if the horizontal lifts of c and c′ with startingpoint p ∈ π−1(u0) have the same end-point in the fibre π−1(u1) above u1.

Equivalently, we can look at the closed loop γ := c (c′)−1 in M , resulting from movingalong the c first and then backwards along c′, and ask, whether parallel transport of p alongγ is the identity or not. As the loop starts and ends in u0, the parallel transport Pγ(p) iscertainly a point in the same fibre π−1(u0) over u0. Since the Lie group G acts transitively onevery fibre, there exists a unique element g := hol(γ) in G with Pγ(p) = rg(p) = p · g.

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This motivates the following definition:

Definition 8.3.1 (Holonomy Group).Let G a finite-dimensional Lie group and E π−→M a principle G-bundle with connection Γ.For every point p ∈ P we define

Hol(Γ, p) := g ∈ G | ∃ closed loop γ around π(p) inMs.t. Pγ(p) = p · g

Hol(Γ, p) is a subgroup of G, called the holonomy group of Γ at p.

Furthermore, we define the restricted holonomy group

Hol0(Γ, p) := g ∈ Hol(Γ, p) | the corresponding γ can be chosen to be null-homotopic

by restricting to null-homotopic curves in M .

In our case, the base-manifold Ures(H) or GLres(H), respectively, is simply-connected andtherefore Hol and Hol0 are the same.

It is easy to check that the holonomy group is indeed a group. Concatenation of two loopsresults in multiplication of the corresponding group elements, parallel transport along the con-stant path yields the identity and the inverse of a group element is obtained by reversing thesense of the corresponding path, i.e. moving along that path backwards.

More formally, if p ∈ E, I = [0, 1] ⊂ R and γ : I →M denotes closed loops around π(p) ∈M ,we have

I) Pγ0(p) = p, for the constant path γ0(t) ≡ π(p).

II) Pγ(p) = p · g ⇒ Pγ−1(p) = p · g−1, for γ−1 : I →M defined by γ−1(t) = γ(1− t).

III) Pγi(p) = p · gi ⇒ Pγ1·γ2(p) = p · (g1g2), where γ1 · γ2(t) =

γ1(2t), t ∈ [0, 1

2 ]

γ2(2t− 1), t ∈ [ 12 , 1]

.

The holonomy groups at two points which can be connected by parallel transport are conju-gated to each other because, if p, q can be connected by a horizontal curve in E, we can paralleltransport from q to p, then along a loop around p and back from p to q, which corresponds toparallel transport along a loop around q.

So much for the basics. Now we are going to show that the holonomy group of the bundleUres(H)

π−→ Ures(H) at the identity is homomorphic to U(1), i.e. to the entire structure group.We could perform the same calculation for the GLres-bundle, but as we are mainly interestedin lifting paths in Ures(H) ⊂ GLres(H), we find the unitary case more educative.

It will actually suffice to consider loops in a two-dimensional subspace. Thus, we will dothe computations for U(2,C) for simplicity and embed U(2,C) into Ures(H) in the followingway: Let (ek)k∈Z be a basis of H such that (ek)k≥0 is a basis of H+ and (ek)k<0 a basis of H−.

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Now can identify U(2) with U(span(e0, e−1)) i.e.

U(2,C) → Ures(H);

(a b

c d

)7−→

a b

1 0c d

0 1

,

where the identity matrices are on (e0)⊥ ⊂ H+ and (e−1)⊥ ⊂ H−, respectively.

A general (piecewise differentiable) path in U(2) has the form

U(t) =

(a(t) b(t)

c(t) d(t)

)

with

U−1(t) = U∗(t) =

(α(t) β(t)

γ(t) δ(t)

)=

(a∗(t) c∗(t)

b∗(t) d∗(t)

).

Unitarity requires (among other identitites) |a|2 + |b|2 = 1.The formula (7.2.1) for the parallel transport expressed in local coordinates becomes

exp[ T∫−T

tr[a(t)(α(t)− a−1(t)) + b(t)γ(t)

]dt]

= exp[ T∫−T

[a(t)(a∗(t)− a−1(t)) + b(t)b∗(t)

]dt].

(8.3.1)We can write

a(t) = r(t) eiϕ(t)

b(t) =√

1− r2(t) eiψ(t)

with r(t), ϕ(t) and ψ(t) piecewise differentiable, real functions. Our path has to stay in theneighborhood of 1 where a(t) is invertible, i.e. r(t) 6= 0 is required. Note that |r(t)| ≤ 1, ∀t,so√

1− r2(t) is a real, differentiable function. Actually, we won’t even have to exploit thefreedom of choosing ψ(t) and can set it to zero. Then we compute:

a(t) = r(t)eiϕ(t) + iϕ(t)r(t)eiϕ(t)

a(t)a∗(t) = r(t)r(t) + iϕ(t)r2(t)

a(t)a−1(t) = r(t)r−1(t) + iϕ(t)

b(t)b∗(t) = −r(t)r(t)

The argument of the exponential in (8.3.1) is thus∫ [a(t)a∗(t)− a(t)a−1(t) + b(t)b∗(t)

]dt

=

∫ [i ϕ(t)(r2(t)− 1)− r−1(t)r(t)

]dt. (8.3.2)

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The second summand is just the derivative of log(r(t)

)and gives no contribution when inte-

grated over a closed loop. Thus, we’re left with

exp[∮ [

i ϕ(t)(r2(t)− 1)− r−1(t)r(t)]dt]

= exp[i

∮ [ϕ(t)(r2(t)− 1)

]dt], (8.3.3)

where the integral is over the parameterization of a closed path around the identity in U(2),corresponding to the boundary conditions r(−T ) = r(+T ) = 1 and ϕ(±T ) ∈ 2πZ.

But (8.3.3) can take any value in U(1): We parameterize over t ∈ [0, 1] and set ϕ(t) := 2πt, suchthat ϕ(t) ≡ 2π. Now, for any δ ∈ [0, 1) we can choose a smooth cut-off function ρ with compactsupport in [0, 1] and values in [0, 1), such that

∫ρ(t) dt = δ. Let r(t) :=

√1− ρ(t), t ∈ [0, 1].

r is smooth and strictly positive with r(0) = r(1) = 1. For this, we compute

1∫0

[ϕ(t)(r2(t)− 1)

]dt = 2π

1∫0

ρ(t) dt = 2πδ

which can take any value between 0 and 2π, including 0.

So, we have found that parallel transport along a closed path (about the identity) can result inmultiplication by any complex phase. The holonomy group is the whole U(1). An analogouscomputation reveals that the holonomy group of the GLres(H)-bundle, is also homomorphic toits entire structure group C×.

We summarize:

Proposition 8.3.2 (Holonomy Groups).The holonomy groups of the connection ΓΘ on the principle bundle GLres(H) and of its restric-tion to Ures(H) are

Hol(GLres,ΓΘ) = Hol0(GLres,ΓΘ) = C×

andHol(Ures,ΓΘ) = Hol0(Ures,ΓΘ) = U(1).

For us, this means that the entire freedom of the geometric phase that we had eliminated byparallel transport, is actually reintroduced through the ambiguity in the choice of the renor-malization. It merely gets a new name: holonomy.

We remark that on a finite-dimensional bundle we could have immediately derived this re-sult by means of the Ambrose-Singer Theorem which states that the Lie algebra of the holonomygroup is generated by the curvature two-form of the connection (it would suffice to note thatthe Schwinger cocycle evaluated on ures × ures can take any value in iR = Lie(U(1))). Unfor-tunately, generalization to infinite-dimensions is usually problematic, since the proof relies onthe Frobenius theorem which has no equipollent infinite-dimensional analogon. Therefore, weprefer to avoid all complications by the direct computation performed above.

123


Holonomy: explicit formula

There is an explicit formula to compute how the parallel transport along two curves differ. Incoordinates defined by a local section σ, the holonomy-group element hol(γ) corresponding toparallel transport along a closed loop γ w.r.to the connection one-form Θ can be computed as

hol(γ) = exp[∮γ

(σ∗Θ)]

= exp[ ∫S(γ)

(σ∗Ω)], (8.3.4)

where S(γ) is the surface enclosed by γ and Ω = dΘ is the curvature 2-form. The first equalityfollows immediately from the local expression for parallel transport, whereas the second equal-ity is an application of Stokes theorem using dΘ = Ω.Note that application of this formula is unproblematic even on infinite-dimensional manifolds,because integration is just along 1-dimensional curves or 2-dimensional surfaces.

If we use two different renormalizations T and T′ to lift the S-matrix by parallel transportalong a path γ = γ[T] as in (8.2.2) we will find that

S[A,T′] = hol(γ[T′] · γ[T]−1) S[A,T] (8.3.5)

and the phase-difference can be computed from (8.3.4).

Remark 8.3.3 (Mickelsson ’98).In [LaMi96], the effects of holonomy are discussed only in connection with gauge transfor-mations, not related to the problem of non-uniqueness of the renormalization. However, J.Mickelsson adresses the issue in a later publication [Mi98], where he suggests that the sec-ond quantization of the S-matrix can be made gauge- and renormalization-independent byintroducing suitable counterterms. Those counterterms can be viewed either as an additionalmodification of the renormalized Hamiltonian, or as choosing a different connection one-form,depending on the A-field. More precisely, it is proven that the phase of the second quantizedscattering matrix, defined by parallel transport using the modified Hamiltonian (respectivelythe modified connection), is invariant under infinitesimal (chirially even) variations of therenormalization 8.1.4 as well as under infinitesimal gauge transformations. To me, the pro-posed solution seems very technically involved and rather ad-hoc. In particular, though, theconstruction violates causality in the sense discussed above, because the counterterms intro-duced by Mickelsson depend explicitly on the entire time evolution UA

I (t,−∞).In the following section, we propose a different, much simpler approach. We are not going

to solve both problems, the gauge-dependence and the renormalization-dependence of theparallel transport. Instead, we are going to argue that we can use the second freedom to ouradvantage and make the geometric second quantization gauge-invariant by choice of suitablerenormalizations.

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8.4. GAUGE INVARIANCE

8.4 Gauge Invariance

We have already seen that gauge transformations are not implementable on the Fock space.Actually, we knew that this must be true, because a gauge transformation

G 3 g : Ψ(x)→ eiΛg(x)Ψ(x), Λg ∈ C∞c (R3,R)

changes the spatial component of the A-field and therefore the polarization class.This fact is troubling, but not necessarily a disaster. It just tells us that we have to give upthe naive idea about gauge invariance (if we ever had it in the first place) that the symmetrytranslates directly from the one-particle theory to the second quantized theory. At this level ofdescription, the significance of a gauge transformation in the second quantized theory is ratherunclear, anyways. We would really have to understand what quantities in the theoreticaldescription have actual physical significance and are in fact required to be gauge-invariant.However, such a discussion, in its full depth, is certainly beyond the scope of this work. Yet,there is little controversy about the fact that the scattering matrix can be taken seriouslyfor the physical description, and therefore, we should require that its gauge invariance carriesover from the one-particle theory to the second quantized theory. This is indeed of particularimportance, because then:

S[Aµ] = S[Aµ − ε∂µΛ], ∀Λ ∈ C∞c (R4,R) (8.4.1)

implies, after taking the derivative with respect to ε at ε = 0:

0 = −∫

dxδ

δAµ(x)S[A] ∂µΛ =

∫dxΛ(x)∂µ

δ

δAµ(x)S[A] (8.4.2)

and thus, by definition (8.0.2) of the current density:

∂µδS

δAµ(x)= ∂µ j

µ(x) ≡ 0. (8.4.3)

So, the gauge invariance of the second quantized S-matrix is physically significant becauseit implies the continuity equation (8.4.3) for the current density. (Actually, equation (8.0.4)shows that a much weaker condition would suffice. The current density will be gauge-invariantif the first distributional derivative δϕ

δA(x) of the phase of S is gauge-invariant. In that case, thecontinuity equation follows analogously from jµ[Aµ] = jµ[Aµ − ε∂µΛ].)

Our method of geometric second quantization is – a priori – not gauge-invariant. Althoughthe one-particle S-operator is invariant under compactly supported gauge transformations, theunitary time evolution and the renormalization are not. Therefore, if we use parallel transportto lift the S-matrix to Ures(H), once for the external field A = (Aµ)µ=0,1,2,3 ∈ C∞c (R4,R4)

and once for the gauge-transformed field A′ = Aµ − ∂µΛ, Λ ∈ C∞c (R4,R), we will perform theparallel transport along different paths in Ures(H) and again, the lifted S-matrix can differ byany complex phase. (This gauge-anomaly can be explicitly computed using formula (8.3.4).See also the explicit computation in [LaMi96].)

125

8.4. GAUGE INVARIANCE

However, we suggest that it is possible to define the renormalization precisely in such a waythat the gauge transformation of the renormalization and the gauge transformation of the timeevolution cancel each other out. We will call such a renormalization gauge-covariant.

Construction 8.4.1 (Gauge-Covariant Renormalization).Let T : R × A → U(H) be a smoothening renormalization, e.g. the renormalization (8.1.4),constructed by Langmann and Mickelsson. Let A be the space of vector potentials and A/Gthe set of gauge-classes, i.e. 4-vector potentials modula the equivalence relation

Aµ ∼ Aµ − ∂µΛ(x), for Λ ∈ C∞c (R4,R). (8.4.4)

More geometrically, we can identify A with the space Ω1(M,R) of (compactly supported) one-forms on Minkowski-space (or a general space-time manifold M). Then, A/G corresponds tothe first de-Rham cohomology class H1

dR(M,R).

By axiom of choice, we choose one representative A out of each gauge-class [A] ∈ A/G. Inparticular, we choose A ≡ 0 as a representative of 0 ∈ A/G. Now, we define a renormalizationT : R× A→ U(H) by setting Tt

(A)

= Tt(A)and

Tt(Aµ − ∂µΛ

):= eiΛ(t,x) Tt

(A), ∀Λ ∈ C∞c (R4,R). (8.4.5)

By theorem 6.2.2, this is indeed a renormalization. Now, with this setting, we find that therenormalized (Schrödinger picture) time evolution is in fact gauge-invariant:

UAren(t, t′) = T

∗t (A) UA(t, t′) T(A(t′))

g ∈ G−−−−−→ UA−∂Λren (t, t′) = T

∗t

(A− ∂µΛ

)UA−∂Λ(t, t′) T

′t

(A− ∂µΛ

)= T

∗t (A) e−iΛ(x) UA−∂Λ(t, t′) eiΛ(x) Tt(A)

= T∗t (A) UA(t, t′) Tt(A) = UA

ren(t, t′)

and because the renormalized time evolution for the distinguished representative is (afterswitching to the interaction picture) differentiable in t with respect to the differentiable struc-ture on Ures(H), the renormalization T is also smoothening.

Using such a gauge-covariant renormalization, the second quantization of the S-matrix iscompletely invariant under gauge transformations – and so is the second quantization of thetime evolution for intermediate times. De facto, we do not see gauge transformations in thesecond quantized theory at all. In particular, the renormalization acts by “gauging the fieldaway” whenever this is possible, which seems like a very elegant solution.

To summarize, we suggest that the problem of the gauge-anomaly for the second quan-tization of the S-matrix by parallel transport, as discussed in [LaMi96] and [Mi98], is notnecessarily a problem at all. Since the choice of a renormalization is ambiguous anyways, wecan use this freedom to cancel out the effect of gauge transformations entirely. In fact, we canjust choose one particular vector potential out of each gauge-class. This choice, however, aswell as the smoothing renormalization applied to it, remains ambiguous.

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Chapter 9

Closing Arguments

What was done.

We have present several equivalent constructions of the fermionic Fock space. In all cases,we arrive at the Shale-Stinespring criterion (1.2.10) as a necessary and sufficient condition fora unitary transformation to be implementable on the Fock space. In this case, the secondquantization is unique up to a complex phase of modulus one, called the “geometric phase” inQED (because it corresponds to the freedom in lifting elements from Ures(H) to the principle-U(1)-bundle Ures(H)). It turns out that the Dirac time evolution does in general not satisfythe Shale-Stinespring condition and thus cannot be second quantized in the usual way. In fact,the time evolution will “leave” the Fock space as soon as the spatial component of the externalfield becomes non-zero (Thm. 1.2.1, Ruijsenaar, Thm. 6.1.1, Deckert et.al.). Physically, thiscorresponds to the problem of infinite particle creation in the presence of a magnetic field. Inother, more technical terms, it means that the free vacuum and the vacua of an interactingtheory correspond to different, non-equivalent representations of the CAR-algebra. Note thatthose are intrinsic properties of the mathematical structure and not pathologies due to theproblem of self-interaction and ultra-violet divergences appearing in the full theory.

The situation is better in the asymptotic case, if we study the S-matrix only. Before theinteraction is turned on and after it is turned off, we have a more or less canonical constructionof the Fock space corresponding to the usual Fock representation and we can compare “instates” (at t = −∞) and “out states” (at t = +∞) with respect to the same vacuum.

We emphasize that these problems, that were discussed here in the context of externalfield Quantum Electrodynamics, are certainly not resolved in the “full-blown” theory (wherethe photon-field is quantized and treated as another dynamic variable), nor are they restrictedto QED or electromagnetic interactions. In fact, a corresponding result is known as Haag’stheorem in algebraic quantum field theory ever since the mid 50’s (see e.g. [StWigh] for a nicetreatment). In the words of A.S. Wightman:

“Not only do strange representations occur ... but different strange representationsat very different time. The defining equations of the interaction picture ... are aswrong as they can possibly be. [...] The strange representations associated withHaag’s theorem are, in fact, an entirely elementary phenomenon and appear assoon as a theory is euclidean invariant and has a Hamiltonian which does nothave the no-particle state as proper vector. This will happen whether or not thetheory is relativistically invariant and whether or not there are ultra-violet problemsin the theory.” [Wigh64], S. 255.

127

After acknowledging those brute and troubling facts, we have presented two different so-lutions for realizing a time evolution in the external field setting of QED. In a way, the twoalternatives seem to be the best that can be done within the existing framework. In Chapter 6,we followed Deckert et.al. and realized the time evolution as unitary transformations betweentime-varying Fock spaces. In Chapter 8, we introduced the concept of a “renormalization” usedto render the time evolution implementable on the standard Fock space. The nomenclature,borrowed from [LaMi96], can be misleading, as it doesn’t refer to the usual renormalizationschemes applied in perturbative field theory but merely to a simple and well-defined pre-scription for mapping the time evolution back into Ures(H).1 In both cases, whether we userenormalizations or time-varying Fock spaces, the implementations of the unitary operatorsare again unique up to a complex phase. We have proven that the two descriptions are dualto each other and showed how the renormalization can be used to translate between them.One advantage of the renormalized theory might be that it allows a second quantization of theHamiltonians as well, after a suitable (“smoothening”) renormalization is applied. However,the renormalization introduces a bunch of artificial terms into the renormalized Hamiltonian(8.1.14), and the physical relevance of these objects remains unclear.

In Chapter 7, we introduced the “Langmann-Mickelsson connection” on the principle bundleUres(H), or rather its complexification GLres(H)→ GLres(H). Parallel transport with respectto this connection defines a unique second quantization of a continuous family of unitaryoperators, including a smooth prescription for the phase. It also defines a unique secondquantization of the renormalized Hamiltonians, generating the second quantized time evolution.However, the hope that the additional geometric structure of a bundle connection can eliminatethe U(1)-freedom of the geometric phase completely, at least for the second quantization ofthe scattering matrix, did not stand up to scrutiny. The reason is that if we want to applythe method of parallel transport to the unitary time evolution, we will always (except for themost trivial cases) need a smoothening renormalization in the sense of Def. 8.1.3 to make thetime evolution differentiable with respect to the differentiable structure on GL1(H+)res(H).The choice of such a renormalization, however, is not unique and different choices will lead todifferent renormalized time evolutions. Although the scattering matrix remains invariant underrenormalization, different choices will correspond to parallel transport along different paths inUres(H) which can lead to different phases for the second quantized scattering operator. Thepath-dependence of the phase is expressed by the holonomy group of the Ures(H) bundle whichwe computed to be isomorphic the the whole structure group U(1). In this sense, the entirefreedom of the geometric phase, supposedly eliminated by the parallel transport, reappears indifferent disguise. We have to conclude that the result stated in [LaMi96], saying that “thephase [of the second quantized scattering operator] is uniquely determined ... by the geometricstructure of the central extension of the group of one-particle (renormalized) time evolutionoperators” is too optimistic. We emphasize that the authors themselves have revised theirinitial statement and addressed the problem in [Mi98].

However, we have shown that this method of geometric second quantization has otherimportant benefits. In Chapter 8 we proved that second quantization by parallel transportpreserves the semi-group structure of the time evolution and – more importantly – the causalstructure of the one-particle theory. In particular, the phase of the second quantized S-matrix

1If and how the two concepts are related might be an interesting question to pursue in a futureanalysis.

128

is causal in the sense of Scharf ([Scha], §2.8).2 The results can be applied in the context oftime-varying Fock spaces, showing that the implementations can be chosen in such a way toyield an actual time evolution. Furthermore, we have proposed a simple solution to the prob-lem of gauge invariance, showing that the geometric second quantization prescription for theS-matrix can be made gauge-invariant by using suitable, “gauge-covariant” renormalizations.Those were the “optimistic” results of our analysis.

Still, we have to realize that at this point, the ambiguities in the construction of the timeevolution seem too vast to provide meaningful physical content. In the framework of time-varying Fock spaces, this fact might be somewhat concealed because the language might suggestthe intuition that the physical state is what it is (modulo phase) and only the mathematicalspace that inhibits it somehow requires additional specification. But in the renormalized theory,this ambiguity translates into the freedom to lift basically any unitary family of operators tothe (fixed) Fock space, as long as it stays in Ures(H) and agrees with the Dirac time evolutionwhenever the interaction is turned off. Therefore, it seems that – without additional structure– the time evolution doesn’t actually tell us anything. In other words: we do not even knowwhat physical quantities should characterize the states represented on different Fock spaces.Without additional ingredients in the theory, specifying (instantaneous) vacuum states, wecannot even say how many particles and anti-particles exist at any given time.3 So, we haveto ask: what actually is the physics behind our mathematical formalism?

What it means.

We concede that – up to this point – our rigorous approaches to the time evolution for theexternal field problem contain too many unspecified degrees of freedom to allow an unambigu-ously defined time evolution and a straight forward physical interpretation in the language ofmany-body Quantum Mechanics. It is an interesting realization on its own – apparent mainlyin the context of time varying Fock spaces – that this problem is closely related to the questionof a consistent particle interpretation. Such a particle interpretation requires a well-definedspecification of (instantaneous) vacuum states that would also define a unique time evolutionon time-varying Fock spaces, up to the geometric phase.

Now, there are basically three general attitudes that can be taken towards this and I’dlike to conclude this work by taking the liberty to discuss them from a rather abstract pointof view. First, we could hope that we can reduce the degrees of freedom – and thereby theambiguities – by introducing additional structure, or that we can impose physical principlesto render the choices involved in our constructions essentially unique. Our analysis shows thatthe principles of causality and gauge invariance are not sufficient to do the job. The questionof Lorentz invariance, however, was addressed only briefly and requires further discussion. Thesituation might also be better, if we restrict the class of permissible interaction potentials, e.g.to actual solutions of Maxwell’s equations. Most likely, though, the crucial insights would haveto come from a fully-interacting theory. We will return to this speculative area later on.

2This feature was already mentioned in [Mi98], however without explanation or proof.3This is one of the big problems of quantum field theory in general. QFT on curved space-time is a

good conscious raiser for this issue. Rather spectacular phenomena like the Unruh effect or Hawkingradiation can be traced back to it.

129

Second, we could conclude that our theory of QED – despite its great success within itslimits – is deeply and fundamentally flawed and that a fundamentally different approach willbe necessary to resolve the various issues. This would certainly be a premature judgement ifbased only on our narrow treatment of the external field problem. But given the fact that allknown version of QED stubbornly resist a consistent mathematical formulation, I personallythink that this alternative has great appeal.

Lastly, one may regard the freedoms contained in the construction not as ambiguities, butrather as a kind of “internal symmetry” of the theory and look for a physical description interms of physical quantities that are invariant under these symmetries, i.e. independent ofthe particular choice of Fock spaces, right-operations etc.. This seems to be more or less theway in which quantum field theories are usually handled, with indisputable practical success.However, I find such an approach very unsatisfying, not because the endeavor is practicallyunpromising, but because it must inevitably lead to the ontological vagueness that I criticizedin the established formulations of quantum field theory. I am convinced that a good physicaltheory must provide a clear physical interpretation on its most fundamental level, preferablyin terms of a clearly specified local ontology.

For example, the results of this work might very well be understood as sustaining thepredominant understanding that not the time evolution but only the S-matrix should be takenseriously in relativistic quantum field theory. But the right question is, whether our failureto do better is a statement about nature, or merely a statement about the capability of ourexisting theories. We can very well conclude with Scharf and Fierz that “the notion of particleshas only asymptotic meaning" ([FiScha79], p. 453). But then, we have to ask, in what termswe are supposed to think about the world as it is right now, i.e. between t = −∞ and t = +∞.If there are no particles, what else IS there?

It is often suggested to be a very deep feature of relativistic quantum theory that physicalrelations can be defined only operationally, i.e. in terms of transition amplitudes, scatteringcross-sections, etc.. (This point of view probably goes back to Heisenberg [Hei43a], [Hei43b],who was convinced that the S-matrix should take over the role of the Hamiltonian in a “future”theory and that all “observable quantities” are encoded in it.) However, one should be scepticalabout the arguments commonly put forward to sustain this claim; quite often, they are eithervery vague or claim a generality which is unjustified, because they are actually trapped withinthe narrow limitations of the current, deficient, framework. In fact, there are some remarkableexamples, demonstrating that a relativistic quantum theory is very well compatible with alocal ontology. There is Tumulkas relativistic version of GRW with flash ontology, for instance,which has caused some furor in recent years (although the theory is not interacting, yet, see[Tum06]). It is also possible – quite easily, in fact – to write down a “Bell-type“ quantum fieldtheory with particle trajectories that reproduces the predictions of regularized QED, includ-ing particle creation and annihilation (although the formulation is not fully Lorentz-invariant,yet, see [DuGTumZ]). Of course, neither of these examples are ultimately satisfying (or evenmeant to be), but they are striking counterexamples to widespread claims, that an understandof relativistic quantum theory in the proper sense cannot be possible.

130

The analysis presented in this work reveals that the scattering matrix is in general a well-defined object in the second quantized theory, whereas the time evolution is not. But it’simportant to note what was actually proven. The one and only reason why things work bet-ter in the asymptotic case is that one assumes the interaction to be turned off in the distantfuture and past (or at least falling off quickly, if the regularity conditions on the fields areweakened). In my opinion, this points to the conclusion that the difficulties are not a matterof Lorentz invariance or some deep feature of relativistic space-time, but that we need a betterunderstanding of the fully interacting theory in terms of the fundamental interactions of itselementary physical entities – presumably the particles.

It was suggested by Dirk Deckert and Detlef Dürr (in talks and private communications)that the Dirac sea theory with infinitely many particles should be understood as an effectivedescription (a thermodynamic limit) of a more fundamental theory, involving a large, but finitenumber of particles with a well-defined pair interaction – either transmitted by the photonfield, or without any photon field at all, like in the Wheeler-Feynman formulation of classicalelectrodynamics [WF45], [WF49]. An analysis of the particle dynamics would then tell us whatthe appropriate “vacuum state” is that we should use in the effective description for specificphysical interactions. It would correspond to an equilibrium state of the “Dirac sea” in whichthe pair interactions within the sea effectively cancel, justifying the description in terms ofDirac’s hole theory (see also the quote of Dirac cited in Chapter 1). At the moment, theseare just speculations, of course. However, I think that this is a path worth pursuing, since itwould provide the clearest and most “down to earth” account for the phenomena of QuantumElectrodynamics.

131

Appendix A

A.1 Commensurability and Polarization ClassesDefinition (Commensurability).Two polarizations V,W ∈ Pol(H) are called commensurable, if V ∩W has finite codimensionin both V and W .

Proposition (Comsurable Polarizations and Polarization Classes).Let V ∈ Pol(H). We denote by Gr(H, V ) the restricted Grassmannian of V i.e. its polarizationclass endowed with the structure of a complex Hilbert-manifold modeled on I2(V, V ⊥). Then,the set of all polarizations W ∈ Pol(H) commensurable with V is dense in Gr(H, V ) and forany such W , charge(V,W ) coincides with (1.2.7), i.e.

charge(V,W ) = ind(PV |W→V ) = dim(V/(V ∩W ))− dim(W/(V ∩W ))

Proof. Let Comns(H,V) := W ∈ Pol(H)|W commensurable with V .

i) Claim: Comns(H,V) ⊂ Gr(H, V ).Let W ∈ Comns(H,V). We have to show PW − PV ∈ I2(H). Obviously, PW − PV is zeroon V ∩W ⊆ H, as well as on (V +W )⊥ ⊆ H. Thus, since V and W are commensurable,i.e. V ∩W has finite codimension in V and W , it has also finite codimension in V + Wand PW − PV is non-zero on a finite-dimensional subspace only and therefore of Hilbert-Schmidt type.

ii) Claim: charge(V,W ) = dim(V/(V ∩W ))− dim(W/(V ∩W )).We can write W = (W ∩ V )⊕ ker(PV |W )⊕R with some subspace R ⊂ H.Then, dim(ker(PV |W )⊕R) <∞ and:

ind(PV |W ) = dim ker(PV |W )− dim coker(PV |W )

= dim(W/(V ∩W ⊕R)

)− dim

(V/(V ∩W ⊕ PV R)

)= dim

(W/(V ∩W )

)− dim

(V/(V ∩W )

)since PV |R is a finite-dimensional isomorphism, hence dim(R) = dim(PVR) <∞.

iii) Claim: Comns(H,V) ⊂ Gr(H, V ) is dense (in the topology introduced in §2.3).Gr(H, V ) is a homogeneous space for Ures(H;V ) := U ∈ U(H) | [PV − PV ⊥ ] ∈ I2(H).Comns(H,V), on the other hand, is the orbit of V under U(H) | [PV−PV ⊥ ] has finite rank .Since the finite-rank operators are dense in the Hilbert-Schmidt class (and the Hilbert-Schmidt norms are smaller than the Ures-norm) it follows that Comns(H,V) is dense inGr(H, V ).

133

A.2. ON BANACH LIE GROUPS

A.2 On Banach Lie groups

Lemma A.2.1 (Continuous One-parameter Groups are smooth).Let A a Banach algebra and G ⊆ A× a linear Lie group.Then, every continuous one-parameter group (γ(t))t∈R in G is smooth.

Proof. Let γ(t), t ∈ R a continuous one-parameter subgroup in G. We write j : G → A× forthe natural inclusion and define γj(t) := j γ(t). Since γ is continuous, there exists ε > 0 with‖γ(t) − 1‖ < 1

2 , ∀t ∈ R, |t| < ε. Let ρ : R → R+ be a smooth cut-off function with compactsupport in [−ε, ε] and

∫Rρ(s)ds = 1.

In a Banach space (even in infinite dimensions) we can do path-integration and thus define

γ(t) := ρ ∗ γj(t) =

∫R

ρ(s)γj(t− s)ds = γx(t)

∫R

ρ(s)γj(−s)ds

= γj(t)

ε∫ε

ρ(s)γj(−s)ds =: γj(t) · g

with g :=ε∫ε

ρ(s)γj(−s)ds ∈ A. We can estimate:

‖g − 1‖ =‖ε∫ε

ρ(s)γx(−s)ds− 1‖ ≤ε∫ε

ρ(s)|γx(−s)− 1‖ds ≤ 1

2

ε∫ε

ρ(s)ds =1

2

so that g ∈ A× i.e. g is invertible. Therefore: γj(t) := γ(t) · g−1. But γ(t) is smooth, since

dk

dtkγ(t) =

dk

dtk

∫R

ρ(s)γx(t− s)ds =dk

dtk

∫R

ρ(t− s)γx(s)ds =

∫R

ρ(k)(t− s)γx(s)ds.

Therefore, γj = γ · g−1 is smooth and so is γ(t).

Proposition A.2.2 (Functorial property of the Lie Algebras).Let G1, G2 be linear Banach Lie groups with Lie algebras Lie(G1) and Lie(G2) , respectively.Let ϕ : G1 → G2 a continuous group homomorphism. Then, the derivative

d

dt

∣∣t=0

ϕ(exp(tx)) =: Lie(ϕ)x

exists for every x ∈ Lie(G1). This defines the unique (continuous) Lie algebra homomorphismLie(ϕ) such that the following diagram commutes:

G1ϕ // G2

Lie(G1)

exp

OO

Lie(ϕ) // Lie(G2)

exp

OO

Proof. Let x ∈ Lie(G1). Then, ϕ(exp(tx)) is a continuous one-parameter group in G2 andtherefore, according to the previous Lemma, even differentiable, i.e. d

dt

∣∣t=0

ϕ(exp(tx)) existsand we define this to be Lie(ϕ)x. But the unique one-parameter group in G2 whose differentialat the identity equals Lie(ϕ)x is t 7→ exp(tLie(ϕ)x). We conclude:

ϕ expG1(x) = expG2

Lie(ϕ)(x), ∀x ∈ Lie(G1). (A.2.1)

Obviously, Lie(ϕ) is R-, respectively C- homogeneous.

134

A.2. ON BANACH LIE GROUPS

To show that it defines a Lie algebra homomorphism, we use the Trotter product formula:

limk→∞

(exp(x/k) exp(y/k)

)k= exp(x+ y) (A.2.2)

and the commutator formula

limk→∞

(exp(x/k) exp(y/k)− exp(y/k) exp(x/k)

)k= exp(xy − yx). (A.2.3)

Then:

ϕ(exp(x+ y)) = limk→∞

ϕ(exp(x/k) exp(y/k)

)k= limk→∞

ϕ(exp(x/k)

)kϕ(exp(y/k)

)k= limk→∞

exp(1

kLie(ϕ)x)

)kexp(

1

kLie(ϕ)y)

)k= exp

(Lie(ϕ)x+ Lie(ϕ)y

).

which meansLie(ϕ)(x+ y) = Lie(ϕ)x+ Lie(ϕ)y.

Similarly, with (A.2.3), ϕ(exp([x, y]) = exp([Lie(ϕ)x,Lie(ϕ)y]

)and thus

Lie(ϕ)([x, y]) =[Lie(ϕ)x,Lie(ϕ)y

].

Hence Lie(ϕ) defines a Lie algebra homomorphism. From (A.2.1) and the fact that the expo-nential map is a local diffeomorphism around 0, it follows that Lie(ϕ) is continuous.

135

A.3. DERIVATION OF THE CHARGE CONJUGATION

A.3 Derivation of the Charge ConjugationWe present a general derivation of the charge conjugation operator C mapping negativeenergy solutions of the Dirac equation to positiv energy solutions with opposite charge.

More precisely, we want a transformation C satisfying

i) CH(e)C−1 = −H(−e)

ii) i~ ∂∂tΨ = H(e)Ψ ⇐⇒ i~ ∂

∂tCΨ = H(−e)CΨ

for H(e) the Dirac-Hamiltonian with charge e and any eigenstate Ψ.i) and ii) imply

i~∂

∂tCΨ = −CH(e)Ψ

For an eigenstate Ψ 6= 0 this is only possible, if C is anti-linear. We can thus write

CΨ = CΨcc.

(cc. denotes complex conjugation).

Now we take a look at the Hamiltonian:

H(e) = −iα · ∇ − eα ·A+mβ + eΦ

and observe that in order to satisfy i), we need

βC = −Cβcc. ; αk C = C αcc.k (A.3.1)

This, together with the (anti-) commutation relations for the α matrices (or γ matrices, re-spectively) are enough to determine the form of the charge conjugation operator in any givenrepresentation. In particular, note that

γi, αj = 0, for i = 0 or i = j[γi, αj

]= 0, else

We look at the two most common examples:

Standard Representation: Only α2 (and thus γ2) is imaginary, all the other matrices arereal. Thus C = const. γ2. Conventionally: C = iγ2 = iβα2

Standard Representation: All the γ’s are imaginary. In particular, β is purely imaginaryand all the α−matrices are real. Hence, looking at (A.3.1), we see that we can take C = 1, i.e.charge-conjugation is just complex conjugation.

136

A.4. MISCELLANEOUS

A.4 Miscellaneous

Lemma A.4.1 (For use in (5.1.21)).For all U ∈ Ures(H) it is true that

dim ker(U++) = dim ker(U∗−−)

dim ker(U−−) = dim ker(U∗++)

Proof. With respect to the splitting H = H+ ⊕ H−, we write U = Ueven + Uodd, with Ueven

the diagonal parts and Uodd the off-diagonal parts. UU∗ = U∗U = 1 then implies

i) (U∗U)even = U∗evenUeven + U∗oddUodd = 1

ii) (U∗U)odd = U∗evenUodd + U∗oddUeven = 0

iii) (UU∗)even = UevenU∗even + UoddU

∗odd = 1

iv) (UU∗)odd = UevenU∗odd + UoddU

∗even = 0

ii) implies Uodd ker(Ueven) ⊂ ker(U∗even)

since U∗evenUodd x = −U∗oddUeven x = 0, ∀x ∈ ker(Ueven).

Similarly, iv) implies U∗odd ker(U∗even) ⊂ ker(Ueven).

We conclude:

Uodd ker(Ueven) ⊂ ker(U∗even)iii)= UoddU

∗odd ker(U∗even) ⊂ Uodd ker(Ueven)

And thusU ker(Ueven) = Uodd ker(Ueven) = ker(U∗even)

Separating w.r.to the ±−splitting yields

U ker(U++) = ker(U∗−−) and U ker(U−−) = ker(U∗++) (A.4.1)

which implies the claimed identities.

137

A.4. MISCELLANEOUS

Norm Estimates for the Dyson Series, Lemma 8.1.7

For the terms in the Dyson series (8.1.13) we proove the norm-estimates [Lem. 8.1.7]:

‖Un(t, t′)‖ ≤ 1

n!

( t∫t′

‖V (s)‖ds

)n, ∀n ≥ 0

and

‖[ε, U1(t, t′)]‖2 ≤t∫

t′

‖[ε, V (s)]‖2 ds

‖[ε, Un(t, t′)]‖2 ≤1

(n− 2)!

t∫t′

‖[ε, V (s)]‖2 ds

( t∫t′

‖V (r)‖dr

)n−1

, ∀n ≥ 1

Proof. Obviously, ‖U0(t, t′)‖ = ‖1‖ = 1, ∀t, t′. And for n ≥ 1:

Un(t, t′) = (−i)nt∫

t′

VI(s1)

s1∫t′

VI(s2) . . .

sn−1∫t′

VI(sn) ds1 . . . dsn

⇒ ‖Un(t, t′)‖ ≤t∫

t′

‖VI(s1)‖s1∫t′

‖VI(s2)‖ . . .sn−1∫t′

‖VI(sn)‖ ds1 . . . dsn

=

t∫t′

‖V (s1)‖s1∫t′

‖V (s2)‖ . . .sn−1∫t′

‖V (sn)‖ ds1 . . . dsn

=1

n!

( t∫t′

‖V (s)‖ds)n

For the I2-estimates we first note that conjugation with eiD0t doesn’t change the Hilbert-Schmidt norm of the odd parts. Thus:

‖[ε, U1(t, t′)]‖2 = ‖t∫

t′

[ε, VI(s)]ds ‖2 ≤t∫

t′

‖[ε, VI(s)]‖2 ds =

t∫t′

‖[ε, V (s)]‖2 ds

Furthermore,

‖[ε, Un+1(t, t′)]‖2 ≤t∫

t′

‖[ε, VI(s)Un(s, t′)]‖2 ds

≤t∫

t′

‖[ε, VI(s)]Un(s, t′)‖2 ds+

t∫t′

‖VI(s)[ε, Un(s, t′)]‖2 ds

≤t∫

t′

‖[ε, VI(s)]‖2 ‖Un(s, t′)‖ ds+

t∫t′

‖VI(s)‖ ‖[ε, Un(s, t′)]‖2 ds

=

t∫t′

‖[ε, V (s)]‖2 ‖Un(s, t′)‖ ds+

t∫t′

‖V (s)‖ ‖[ε, Un(s, t′)]‖2 ds

138

A.4. MISCELLANEOUS

For n=1 this yields

‖[ε, U2(t, t′)]‖2 ≤t∫

t′

s∫t′

(‖[ε, V (s)]‖2 ‖V (r)‖+ ‖V (s)‖ ‖[ε, V (r)]‖2

)dr ds

=1

2

t∫t′

t∫t′

(‖[ε, V (s)]‖2 ‖V (r)‖+ ‖V (s)‖ ‖[ε, V (r)]‖2

)dr ds

=

t∫t′

‖[ε, V (s)]‖2 ds

t∫t′

‖V (s)‖ ds

And thus, inductively, for n ≥ 2

‖[ε, Un+1(t, t′)]‖2 ≤t∫

t′

‖[ε, V (s)]‖2 ‖Un(s, t′)‖ ds+

t∫t′

‖V (s)‖ ‖[ε, Un(s, t′)]‖2 ds

≤ 1

n!

t∫t′

‖[ε, V (s)]‖2( s∫t′

‖V (r)‖dr)n

ds+1

(n− 2)!

t∫t′

‖V (s)‖s∫

t′

‖[ε, V (s′)]‖2 ds′( s∫t′

‖V (r)‖dr)n−1

ds

≤ 1

n!

t∫t′

‖[ε, V (s)]‖2 ds( t∫t′

‖V (r)‖dr)n

+

t∫t′

‖[ε, V (s)]‖2 ds(n− 1)

(n− 1)!

t∫t′

‖V (s)‖( s∫t′

‖V (r)‖dr)n−1

ds

=1

n!

t∫t′

‖[ε, V (s)]‖2 ds( t∫t′

‖V (r)‖dr)n

+(n− 1)

n!

t∫t′

‖[ε, V (s)]‖2 ds( t∫t′

‖V (r)‖dr)n

=1

(n− 1)!

t∫t′

‖[ε, V (s)]‖2 ds( t∫t′

‖V (r)‖dr)n

139

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142

SelbstständigkeitserklärungHiermit erkläre ich, dass ich die vorliegende Arbeit selbstständig und ohneBenutzung anderer als der von mir angegebenen Quellen und Hilfsmittelverfasst habe.

Academic Honor Principle

I hereby certify that I have completed the present thesis independently andwithout the use of sources other than those stated in the thesis.

Dustin LazaroviciMünchen, 30. September 2011

Danksagung Acknowledgement

Ich danke meinen Eltern, Doriana und Dan Lazarovici, die mir mein Studiumermöglicht haben, für ihre Liebe und ihre Unterstützung und für ihre Geduld.

Ich danke meinem Bruder Remy dafür, dass ich immer auf ihn zählen konnte.

Ich danke meinem Lehrer, Detlef Dürr, von dem ich mehr gelernt habe, alsman von irgendeinem Professor erwarten kann. Er hat mir meinen Glaubenbewahrt, dass ein physikalisches Verständins der Welt, im eigentlich Sinne, tat-sächlich möglich ist.

Ich danke Peter Pickl für die Betreuung der Arbeit und dafür, dass er immerdie richtigen, das heißt, physikalischen, Fragen stellt.

Ich danke Dirk Deckert und Franz Merkl für wertvolle Gespräche und kom-petenten Rat. Ohne sie hätte ich oft nicht gewusst, was ich eigentlich mache.

Ich danke meinen guten Freundinnen Janine Adomeit und Sarina Balkhausenfürs Korrekturlesen. Die verbliebenen Tippfehler – es werden zahlreiche sein –sind allein meiner Schludrigkeit und dem Zeitdruck geschuldet.

Ich danke Irène Lassmann für die Kätzchen – sie weiß schon, was gemeint ist.

Ich danke Ingrid Scherer und Verena Heinemann vom mathematischen Insti-tut der LMU München, ohne die ich im Bürokratiedschungel verhungert wäre.

Dustin Lazarovici

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